Properties

Label 1050.2.g.j.799.2
Level 1050
Weight 2
Character 1050.799
Analytic conductor 8.384
Analytic rank 0
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: \(0\)
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 799.2
Root \(-1.00000i\)
Character \(\chi\) = 1050.799
Dual form 1050.2.g.j.799.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} +1.00000i q^{12} -1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} -1.00000 q^{21} +2.00000i q^{22} -7.00000i q^{23} -1.00000 q^{24} +1.00000 q^{26} +1.00000i q^{27} +1.00000i q^{28} -1.00000 q^{29} +3.00000 q^{31} +1.00000i q^{32} -2.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} -4.00000i q^{38} -1.00000 q^{39} -3.00000 q^{41} -1.00000i q^{42} +1.00000i q^{43} -2.00000 q^{44} +7.00000 q^{46} -12.0000i q^{47} -1.00000i q^{48} -1.00000 q^{49} -1.00000 q^{51} +1.00000i q^{52} -11.0000i q^{53} -1.00000 q^{54} -1.00000 q^{56} +4.00000i q^{57} -1.00000i q^{58} +3.00000 q^{59} +5.00000 q^{61} +3.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -12.0000i q^{67} +1.00000i q^{68} -7.00000 q^{69} +4.00000 q^{71} +1.00000i q^{72} +14.0000i q^{73} +6.00000 q^{74} +4.00000 q^{76} -2.00000i q^{77} -1.00000i q^{78} +2.00000 q^{79} +1.00000 q^{81} -3.00000i q^{82} +3.00000i q^{83} +1.00000 q^{84} -1.00000 q^{86} +1.00000i q^{87} -2.00000i q^{88} -10.0000 q^{89} -1.00000 q^{91} +7.00000i q^{92} -3.00000i q^{93} +12.0000 q^{94} +1.00000 q^{96} -10.0000i q^{97} -1.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + 4q^{11} + 2q^{14} + 2q^{16} - 8q^{19} - 2q^{21} - 2q^{24} + 2q^{26} - 2q^{29} + 6q^{31} + 2q^{34} + 2q^{36} - 2q^{39} - 6q^{41} - 4q^{44} + 14q^{46} - 2q^{49} - 2q^{51} - 2q^{54} - 2q^{56} + 6q^{59} + 10q^{61} - 2q^{64} + 4q^{66} - 14q^{69} + 8q^{71} + 12q^{74} + 8q^{76} + 4q^{79} + 2q^{81} + 2q^{84} - 2q^{86} - 20q^{89} - 2q^{91} + 24q^{94} + 2q^{96} - 4q^{99} + O(q^{100}) \)

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
\(17\) − 1.00000i − 0.242536i −0.992620 0.121268i \(-0.961304\pi\)
0.992620 0.121268i \(-0.0386960\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 2.00000i 0.426401i
\(23\) − 7.00000i − 1.45960i −0.683660 0.729800i \(-0.739613\pi\)
0.683660 0.729800i \(-0.260387\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 1.00000i 0.192450i
\(28\) 1.00000i 0.188982i
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 2.00000i − 0.348155i
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) − 1.00000i − 0.154303i
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 7.00000 1.03209
\(47\) − 12.0000i − 1.75038i −0.483779 0.875190i \(-0.660736\pi\)
0.483779 0.875190i \(-0.339264\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 1.00000i 0.138675i
\(53\) − 11.0000i − 1.51097i −0.655168 0.755483i \(-0.727402\pi\)
0.655168 0.755483i \(-0.272598\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 4.00000i 0.529813i
\(58\) − 1.00000i − 0.131306i
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 3.00000i 0.381000i
\(63\) 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 \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 1.00000i 0.121268i
\(69\) −7.00000 −0.842701
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) − 2.00000i − 0.227921i
\(78\) − 1.00000i − 0.113228i
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 3.00000i − 0.331295i
\(83\) 3.00000i 0.329293i 0.986353 + 0.164646i \(0.0526483\pi\)
−0.986353 + 0.164646i \(0.947352\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) 1.00000i 0.107211i
\(88\) − 2.00000i − 0.213201i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 7.00000i 0.729800i
\(93\) − 3.00000i − 0.311086i
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) − 1.00000i − 0.0990148i
\(103\) 17.0000i 1.67506i 0.546392 + 0.837530i \(0.316001\pi\)
−0.546392 + 0.837530i \(0.683999\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 11.0000 1.06841
\(107\) − 18.0000i − 1.74013i −0.492941 0.870063i \(-0.664078\pi\)
0.492941 0.870063i \(-0.335922\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) − 1.00000i − 0.0944911i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 1.00000 0.0928477
\(117\) 1.00000i 0.0924500i
\(118\) 3.00000i 0.276172i
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 5.00000i 0.452679i
\(123\) 3.00000i 0.270501i
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) 14.0000i 1.24230i 0.783692 + 0.621150i \(0.213334\pi\)
−0.783692 + 0.621150i \(0.786666\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 2.00000i 0.174078i
\(133\) 4.00000i 0.346844i
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) 4.00000i 0.341743i 0.985293 + 0.170872i \(0.0546583\pi\)
−0.985293 + 0.170872i \(0.945342\pi\)
\(138\) − 7.00000i − 0.595880i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 4.00000i 0.335673i
\(143\) − 2.00000i − 0.167248i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) 1.00000i 0.0824786i
\(148\) 6.00000i 0.493197i
\(149\) −5.00000 −0.409616 −0.204808 0.978802i \(-0.565657\pi\)
−0.204808 + 0.978802i \(0.565657\pi\)
\(150\) 0 0
\(151\) 22.0000 1.79033 0.895167 0.445730i \(-0.147056\pi\)
0.895167 + 0.445730i \(0.147056\pi\)
\(152\) 4.00000i 0.324443i
\(153\) 1.00000i 0.0808452i
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) 2.00000i 0.159111i
\(159\) −11.0000 −0.872357
\(160\) 0 0
\(161\) −7.00000 −0.551677
\(162\) 1.00000i 0.0785674i
\(163\) 19.0000i 1.48819i 0.668071 + 0.744097i \(0.267120\pi\)
−0.668071 + 0.744097i \(0.732880\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) −3.00000 −0.232845
\(167\) − 2.00000i − 0.154765i −0.997001 0.0773823i \(-0.975344\pi\)
0.997001 0.0773823i \(-0.0246562\pi\)
\(168\) 1.00000i 0.0771517i
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) − 1.00000i − 0.0762493i
\(173\) 12.0000i 0.912343i 0.889892 + 0.456172i \(0.150780\pi\)
−0.889892 + 0.456172i \(0.849220\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) − 3.00000i − 0.225494i
\(178\) − 10.0000i − 0.749532i
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 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
\(183\) − 5.00000i − 0.369611i
\(184\) −7.00000 −0.516047
\(185\) 0 0
\(186\) 3.00000 0.219971
\(187\) − 2.00000i − 0.146254i
\(188\) 12.0000i 0.875190i
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −13.0000 −0.940647 −0.470323 0.882494i \(-0.655863\pi\)
−0.470323 + 0.882494i \(0.655863\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 27.0000i 1.92367i 0.273629 + 0.961835i \(0.411776\pi\)
−0.273629 + 0.961835i \(0.588224\pi\)
\(198\) − 2.00000i − 0.142134i
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 0 0
\(203\) 1.00000i 0.0701862i
\(204\) 1.00000 0.0700140
\(205\) 0 0
\(206\) −17.0000 −1.18445
\(207\) 7.00000i 0.486534i
\(208\) − 1.00000i − 0.0693375i
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) 15.0000 1.03264 0.516321 0.856395i \(-0.327301\pi\)
0.516321 + 0.856395i \(0.327301\pi\)
\(212\) 11.0000i 0.755483i
\(213\) − 4.00000i − 0.274075i
\(214\) 18.0000 1.23045
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) − 3.00000i − 0.203653i
\(218\) 4.00000i 0.270914i
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −1.00000 −0.0672673
\(222\) − 6.00000i − 0.402694i
\(223\) 13.0000i 0.870544i 0.900299 + 0.435272i \(0.143348\pi\)
−0.900299 + 0.435272i \(0.856652\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 7.00000i 0.464606i 0.972643 + 0.232303i \(0.0746261\pi\)
−0.972643 + 0.232303i \(0.925374\pi\)
\(228\) − 4.00000i − 0.264906i
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 1.00000i 0.0656532i
\(233\) − 20.0000i − 1.31024i −0.755523 0.655122i \(-0.772617\pi\)
0.755523 0.655122i \(-0.227383\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −3.00000 −0.195283
\(237\) − 2.00000i − 0.129914i
\(238\) − 1.00000i − 0.0648204i
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) − 7.00000i − 0.449977i
\(243\) − 1.00000i − 0.0641500i
\(244\) −5.00000 −0.320092
\(245\) 0 0
\(246\) −3.00000 −0.191273
\(247\) 4.00000i 0.254514i
\(248\) − 3.00000i − 0.190500i
\(249\) 3.00000 0.190117
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) − 1.00000i − 0.0629941i
\(253\) − 14.0000i − 0.880172i
\(254\) −14.0000 −0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 5.00000i − 0.311891i −0.987766 0.155946i \(-0.950158\pi\)
0.987766 0.155946i \(-0.0498425\pi\)
\(258\) 1.00000i 0.0622573i
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 8.00000i 0.494242i
\(263\) − 11.0000i − 0.678289i −0.940734 0.339145i \(-0.889862\pi\)
0.940734 0.339145i \(-0.110138\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) 10.0000i 0.611990i
\(268\) 12.0000i 0.733017i
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) − 1.00000i − 0.0606339i
\(273\) 1.00000i 0.0605228i
\(274\) −4.00000 −0.241649
\(275\) 0 0
\(276\) 7.00000 0.421350
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 4.00000i 0.239904i
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) − 12.0000i − 0.714590i
\(283\) 20.0000i 1.18888i 0.804141 + 0.594438i \(0.202626\pi\)
−0.804141 + 0.594438i \(0.797374\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 3.00000i 0.177084i
\(288\) − 1.00000i − 0.0589256i
\(289\) 16.0000 0.941176
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) − 14.0000i − 0.819288i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 2.00000i 0.116052i
\(298\) − 5.00000i − 0.289642i
\(299\) −7.00000 −0.404820
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 22.0000i 1.26596i
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) − 22.0000i − 1.25561i −0.778372 0.627803i \(-0.783954\pi\)
0.778372 0.627803i \(-0.216046\pi\)
\(308\) 2.00000i 0.113961i
\(309\) 17.0000 0.967096
\(310\) 0 0
\(311\) 34.0000 1.92796 0.963982 0.265969i \(-0.0856919\pi\)
0.963982 + 0.265969i \(0.0856919\pi\)
\(312\) 1.00000i 0.0566139i
\(313\) − 18.0000i − 1.01742i −0.860938 0.508710i \(-0.830123\pi\)
0.860938 0.508710i \(-0.169877\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) − 17.0000i − 0.954815i −0.878682 0.477408i \(-0.841577\pi\)
0.878682 0.477408i \(-0.158423\pi\)
\(318\) − 11.0000i − 0.616849i
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) − 7.00000i − 0.390095i
\(323\) 4.00000i 0.222566i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −19.0000 −1.05231
\(327\) − 4.00000i − 0.221201i
\(328\) 3.00000i 0.165647i
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −3.00000 −0.164895 −0.0824475 0.996595i \(-0.526274\pi\)
−0.0824475 + 0.996595i \(0.526274\pi\)
\(332\) − 3.00000i − 0.164646i
\(333\) 6.00000i 0.328798i
\(334\) 2.00000 0.109435
\(335\) 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
\(341\) 6.00000 0.324918
\(342\) 4.00000i 0.216295i
\(343\) 1.00000i 0.0539949i
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) 26.0000i 1.39575i 0.716218 + 0.697877i \(0.245872\pi\)
−0.716218 + 0.697877i \(0.754128\pi\)
\(348\) − 1.00000i − 0.0536056i
\(349\) 1.00000 0.0535288 0.0267644 0.999642i \(-0.491480\pi\)
0.0267644 + 0.999642i \(0.491480\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 2.00000i 0.106600i
\(353\) − 10.0000i − 0.532246i −0.963939 0.266123i \(-0.914257\pi\)
0.963939 0.266123i \(-0.0857428\pi\)
\(354\) 3.00000 0.159448
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 1.00000i 0.0529256i
\(358\) − 10.0000i − 0.528516i
\(359\) 9.00000 0.475002 0.237501 0.971387i \(-0.423672\pi\)
0.237501 + 0.971387i \(0.423672\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 18.0000i − 0.946059i
\(363\) 7.00000i 0.367405i
\(364\) 1.00000 0.0524142
\(365\) 0 0
\(366\) 5.00000 0.261354
\(367\) 13.0000i 0.678594i 0.940679 + 0.339297i \(0.110189\pi\)
−0.940679 + 0.339297i \(0.889811\pi\)
\(368\) − 7.00000i − 0.364900i
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) −11.0000 −0.571092
\(372\) 3.00000i 0.155543i
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 1.00000i 0.0515026i
\(378\) 1.00000i 0.0514344i
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) 0 0
\(381\) 14.0000 0.717242
\(382\) − 13.0000i − 0.665138i
\(383\) 20.0000i 1.02195i 0.859595 + 0.510976i \(0.170716\pi\)
−0.859595 + 0.510976i \(0.829284\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) − 1.00000i − 0.0508329i
\(388\) 10.0000i 0.507673i
\(389\) −22.0000 −1.11544 −0.557722 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 0 0
\(391\) −7.00000 −0.354005
\(392\) 1.00000i 0.0505076i
\(393\) − 8.00000i − 0.403547i
\(394\) −27.0000 −1.36024
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 3.00000i 0.150566i 0.997162 + 0.0752828i \(0.0239860\pi\)
−0.997162 + 0.0752828i \(0.976014\pi\)
\(398\) 24.0000i 1.20301i
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) 16.0000 0.799002 0.399501 0.916733i \(-0.369183\pi\)
0.399501 + 0.916733i \(0.369183\pi\)
\(402\) − 12.0000i − 0.598506i
\(403\) − 3.00000i − 0.149441i
\(404\) 0 0
\(405\) 0 0
\(406\) −1.00000 −0.0496292
\(407\) − 12.0000i − 0.594818i
\(408\) 1.00000i 0.0495074i
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 4.00000 0.197305
\(412\) − 17.0000i − 0.837530i
\(413\) − 3.00000i − 0.147620i
\(414\) −7.00000 −0.344031
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) − 4.00000i − 0.195881i
\(418\) − 8.00000i − 0.391293i
\(419\) 25.0000 1.22133 0.610665 0.791889i \(-0.290902\pi\)
0.610665 + 0.791889i \(0.290902\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 15.0000i 0.730189i
\(423\) 12.0000i 0.583460i
\(424\) −11.0000 −0.534207
\(425\) 0 0
\(426\) 4.00000 0.193801
\(427\) − 5.00000i − 0.241967i
\(428\) 18.0000i 0.870063i
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) 27.0000 1.30054 0.650272 0.759701i \(-0.274655\pi\)
0.650272 + 0.759701i \(0.274655\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 34.0000i − 1.63394i −0.576683 0.816968i \(-0.695653\pi\)
0.576683 0.816968i \(-0.304347\pi\)
\(434\) 3.00000 0.144005
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 28.0000i 1.33942i
\(438\) 14.0000i 0.668946i
\(439\) −7.00000 −0.334092 −0.167046 0.985949i \(-0.553423\pi\)
−0.167046 + 0.985949i \(0.553423\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) − 1.00000i − 0.0475651i
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) −13.0000 −0.615568
\(447\) 5.00000i 0.236492i
\(448\) 1.00000i 0.0472456i
\(449\) 16.0000 0.755087 0.377543 0.925992i \(-0.376769\pi\)
0.377543 + 0.925992i \(0.376769\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 6.00000i 0.282216i
\(453\) − 22.0000i − 1.03365i
\(454\) −7.00000 −0.328526
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) − 17.0000i − 0.795226i −0.917553 0.397613i \(-0.869839\pi\)
0.917553 0.397613i \(-0.130161\pi\)
\(458\) − 14.0000i − 0.654177i
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) 32.0000 1.49039 0.745194 0.666847i \(-0.232357\pi\)
0.745194 + 0.666847i \(0.232357\pi\)
\(462\) − 2.00000i − 0.0930484i
\(463\) − 36.0000i − 1.67306i −0.547920 0.836531i \(-0.684580\pi\)
0.547920 0.836531i \(-0.315420\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) 20.0000 0.926482
\(467\) − 39.0000i − 1.80470i −0.430999 0.902352i \(-0.641839\pi\)
0.430999 0.902352i \(-0.358161\pi\)
\(468\) − 1.00000i − 0.0462250i
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) − 3.00000i − 0.138086i
\(473\) 2.00000i 0.0919601i
\(474\) 2.00000 0.0918630
\(475\) 0 0
\(476\) 1.00000 0.0458349
\(477\) 11.0000i 0.503655i
\(478\) 24.0000i 1.09773i
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) − 26.0000i − 1.18427i
\(483\) 7.00000i 0.318511i
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 22.0000i 0.996915i 0.866914 + 0.498458i \(0.166100\pi\)
−0.866914 + 0.498458i \(0.833900\pi\)
\(488\) − 5.00000i − 0.226339i
\(489\) 19.0000 0.859210
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) − 3.00000i − 0.135250i
\(493\) 1.00000i 0.0450377i
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 3.00000 0.134704
\(497\) − 4.00000i − 0.179425i
\(498\) 3.00000i 0.134433i
\(499\) 27.0000 1.20869 0.604343 0.796724i \(-0.293436\pi\)
0.604343 + 0.796724i \(0.293436\pi\)
\(500\) 0 0
\(501\) −2.00000 −0.0893534
\(502\) − 15.0000i − 0.669483i
\(503\) 2.00000i 0.0891756i 0.999005 + 0.0445878i \(0.0141974\pi\)
−0.999005 + 0.0445878i \(0.985803\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 14.0000 0.622376
\(507\) − 12.0000i − 0.532939i
\(508\) − 14.0000i − 0.621150i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 1.00000i 0.0441942i
\(513\) − 4.00000i − 0.176604i
\(514\) 5.00000 0.220541
\(515\) 0 0
\(516\) −1.00000 −0.0440225
\(517\) − 24.0000i − 1.05552i
\(518\) − 6.00000i − 0.263625i
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) 39.0000 1.70862 0.854311 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) 1.00000i 0.0437688i
\(523\) 8.00000i 0.349816i 0.984585 + 0.174908i \(0.0559627\pi\)
−0.984585 + 0.174908i \(0.944037\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 11.0000 0.479623
\(527\) − 3.00000i − 0.130682i
\(528\) − 2.00000i − 0.0870388i
\(529\) −26.0000 −1.13043
\(530\) 0 0
\(531\) −3.00000 −0.130189
\(532\) − 4.00000i − 0.173422i
\(533\) 3.00000i 0.129944i
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 10.0000i 0.431532i
\(538\) − 18.0000i − 0.776035i
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) −28.0000 −1.20381 −0.601907 0.798566i \(-0.705592\pi\)
−0.601907 + 0.798566i \(0.705592\pi\)
\(542\) − 20.0000i − 0.859074i
\(543\) 18.0000i 0.772454i
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) −1.00000 −0.0427960
\(547\) − 15.0000i − 0.641354i −0.947189 0.320677i \(-0.896090\pi\)
0.947189 0.320677i \(-0.103910\pi\)
\(548\) − 4.00000i − 0.170872i
\(549\) −5.00000 −0.213395
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 7.00000i 0.297940i
\(553\) − 2.00000i − 0.0850487i
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) − 3.00000i − 0.127000i
\(559\) 1.00000 0.0422955
\(560\) 0 0
\(561\) −2.00000 −0.0844401
\(562\) 0 0
\(563\) − 41.0000i − 1.72794i −0.503540 0.863972i \(-0.667969\pi\)
0.503540 0.863972i \(-0.332031\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) − 1.00000i − 0.0419961i
\(568\) − 4.00000i − 0.167836i
\(569\) −14.0000 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) 0 0
\(571\) 19.0000 0.795125 0.397563 0.917575i \(-0.369856\pi\)
0.397563 + 0.917575i \(0.369856\pi\)
\(572\) 2.00000i 0.0836242i
\(573\) 13.0000i 0.543083i
\(574\) −3.00000 −0.125218
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 34.0000i 1.41544i 0.706494 + 0.707719i \(0.250276\pi\)
−0.706494 + 0.707719i \(0.749724\pi\)
\(578\) 16.0000i 0.665512i
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 3.00000 0.124461
\(582\) − 10.0000i − 0.414513i
\(583\) − 22.0000i − 0.911147i
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) 0 0
\(587\) − 27.0000i − 1.11441i −0.830375 0.557205i \(-0.811874\pi\)
0.830375 0.557205i \(-0.188126\pi\)
\(588\) − 1.00000i − 0.0412393i
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 27.0000 1.11063
\(592\) − 6.00000i − 0.246598i
\(593\) 18.0000i 0.739171i 0.929197 + 0.369586i \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 5.00000 0.204808
\(597\) − 24.0000i − 0.982255i
\(598\) − 7.00000i − 0.286251i
\(599\) 45.0000 1.83865 0.919325 0.393499i \(-0.128735\pi\)
0.919325 + 0.393499i \(0.128735\pi\)
\(600\) 0 0
\(601\) 28.0000 1.14214 0.571072 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) 1.00000i 0.0407570i
\(603\) 12.0000i 0.488678i
\(604\) −22.0000 −0.895167
\(605\) 0 0
\(606\) 0 0
\(607\) 28.0000i 1.13648i 0.822861 + 0.568242i \(0.192376\pi\)
−0.822861 + 0.568242i \(0.807624\pi\)
\(608\) − 4.00000i − 0.162221i
\(609\) 1.00000 0.0405220
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) − 1.00000i − 0.0404226i
\(613\) − 18.0000i − 0.727013i −0.931592 0.363507i \(-0.881579\pi\)
0.931592 0.363507i \(-0.118421\pi\)
\(614\) 22.0000 0.887848
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) 14.0000i 0.563619i 0.959470 + 0.281809i \(0.0909346\pi\)
−0.959470 + 0.281809i \(0.909065\pi\)
\(618\) 17.0000i 0.683840i
\(619\) −38.0000 −1.52735 −0.763674 0.645601i \(-0.776607\pi\)
−0.763674 + 0.645601i \(0.776607\pi\)
\(620\) 0 0
\(621\) 7.00000 0.280900
\(622\) 34.0000i 1.36328i
\(623\) 10.0000i 0.400642i
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) 18.0000 0.719425
\(627\) 8.00000i 0.319489i
\(628\) − 18.0000i − 0.718278i
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 26.0000 1.03504 0.517522 0.855670i \(-0.326855\pi\)
0.517522 + 0.855670i \(0.326855\pi\)
\(632\) − 2.00000i − 0.0795557i
\(633\) − 15.0000i − 0.596196i
\(634\) 17.0000 0.675156
\(635\) 0 0
\(636\) 11.0000 0.436178
\(637\) 1.00000i 0.0396214i
\(638\) − 2.00000i − 0.0791808i
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) −26.0000 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) − 18.0000i − 0.710403i
\(643\) − 2.00000i − 0.0788723i −0.999222 0.0394362i \(-0.987444\pi\)
0.999222 0.0394362i \(-0.0125562\pi\)
\(644\) 7.00000 0.275839
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) 6.00000i 0.235884i 0.993020 + 0.117942i \(0.0376297\pi\)
−0.993020 + 0.117942i \(0.962370\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 6.00000 0.235521
\(650\) 0 0
\(651\) −3.00000 −0.117579
\(652\) − 19.0000i − 0.744097i
\(653\) − 14.0000i − 0.547862i −0.961749 0.273931i \(-0.911676\pi\)
0.961749 0.273931i \(-0.0883240\pi\)
\(654\) 4.00000 0.156412
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) − 14.0000i − 0.546192i
\(658\) − 12.0000i − 0.467809i
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) − 3.00000i − 0.116598i
\(663\) 1.00000i 0.0388368i
\(664\) 3.00000 0.116423
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 7.00000i 0.271041i
\(668\) 2.00000i 0.0773823i
\(669\) 13.0000 0.502609
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) − 1.00000i − 0.0385758i
\(673\) − 27.0000i − 1.04077i −0.853931 0.520387i \(-0.825788\pi\)
0.853931 0.520387i \(-0.174212\pi\)
\(674\) −27.0000 −1.04000
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) − 20.0000i − 0.768662i −0.923195 0.384331i \(-0.874432\pi\)
0.923195 0.384331i \(-0.125568\pi\)
\(678\) − 6.00000i − 0.230429i
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 7.00000 0.268241
\(682\) 6.00000i 0.229752i
\(683\) 6.00000i 0.229584i 0.993390 + 0.114792i \(0.0366201\pi\)
−0.993390 + 0.114792i \(0.963380\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 14.0000i 0.534133i
\(688\) 1.00000i 0.0381246i
\(689\) −11.0000 −0.419067
\(690\) 0 0
\(691\) −42.0000 −1.59776 −0.798878 0.601494i \(-0.794573\pi\)
−0.798878 + 0.601494i \(0.794573\pi\)
\(692\) − 12.0000i − 0.456172i
\(693\) 2.00000i 0.0759737i
\(694\) −26.0000 −0.986947
\(695\) 0 0
\(696\) 1.00000 0.0379049
\(697\) 3.00000i 0.113633i
\(698\) 1.00000i 0.0378506i
\(699\) −20.0000 −0.756469
\(700\) 0 0
\(701\) −39.0000 −1.47301 −0.736505 0.676432i \(-0.763525\pi\)
−0.736505 + 0.676432i \(0.763525\pi\)
\(702\) 1.00000i 0.0377426i
\(703\) 24.0000i 0.905177i
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) 0 0
\(708\) 3.00000i 0.112747i
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) −2.00000 −0.0750059
\(712\) 10.0000i 0.374766i
\(713\) − 21.0000i − 0.786456i
\(714\) −1.00000 −0.0374241
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) − 24.0000i − 0.896296i
\(718\) 9.00000i 0.335877i
\(719\) 34.0000 1.26799 0.633993 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(720\) 0 0
\(721\) 17.0000 0.633113
\(722\) − 3.00000i − 0.111648i
\(723\) 26.0000i 0.966950i
\(724\) 18.0000 0.668965
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) − 1.00000i − 0.0370879i −0.999828 0.0185440i \(-0.994097\pi\)
0.999828 0.0185440i \(-0.00590307\pi\)
\(728\) 1.00000i 0.0370625i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 1.00000 0.0369863
\(732\) 5.00000i 0.184805i
\(733\) − 19.0000i − 0.701781i −0.936416 0.350891i \(-0.885879\pi\)
0.936416 0.350891i \(-0.114121\pi\)
\(734\) −13.0000 −0.479839
\(735\) 0 0
\(736\) 7.00000 0.258023
\(737\) − 24.0000i − 0.884051i
\(738\) 3.00000i 0.110432i
\(739\) −11.0000 −0.404642 −0.202321 0.979319i \(-0.564848\pi\)
−0.202321 + 0.979319i \(0.564848\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) − 11.0000i − 0.403823i
\(743\) 3.00000i 0.110059i 0.998485 + 0.0550297i \(0.0175253\pi\)
−0.998485 + 0.0550297i \(0.982475\pi\)
\(744\) −3.00000 −0.109985
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) − 3.00000i − 0.109764i
\(748\) 2.00000i 0.0731272i
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) − 12.0000i − 0.437595i
\(753\) 15.0000i 0.546630i
\(754\) −1.00000 −0.0364179
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) − 2.00000i − 0.0726912i −0.999339 0.0363456i \(-0.988428\pi\)
0.999339 0.0363456i \(-0.0115717\pi\)
\(758\) 5.00000i 0.181608i
\(759\) −14.0000 −0.508168
\(760\) 0 0
\(761\) 34.0000 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) 14.0000i 0.507166i
\(763\) − 4.00000i − 0.144810i
\(764\) 13.0000 0.470323
\(765\) 0 0
\(766\) −20.0000 −0.722629
\(767\) − 3.00000i − 0.108324i
\(768\) − 1.00000i − 0.0360844i
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) −5.00000 −0.180071
\(772\) 2.00000i 0.0719816i
\(773\) 20.0000i 0.719350i 0.933078 + 0.359675i \(0.117112\pi\)
−0.933078 + 0.359675i \(0.882888\pi\)
\(774\) 1.00000 0.0359443
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 6.00000i 0.215249i
\(778\) − 22.0000i − 0.788738i
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) − 7.00000i − 0.250319i
\(783\) − 1.00000i − 0.0357371i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 8.00000 0.285351
\(787\) 38.0000i 1.35455i 0.735728 + 0.677277i \(0.236840\pi\)
−0.735728 + 0.677277i \(0.763160\pi\)
\(788\) − 27.0000i − 0.961835i
\(789\) −11.0000 −0.391610
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 2.00000i 0.0710669i
\(793\) − 5.00000i − 0.177555i
\(794\) −3.00000 −0.106466
\(795\) 0 0
\(796\) −24.0000 −0.850657
\(797\) 42.0000i 1.48772i 0.668338 + 0.743858i \(0.267006\pi\)
−0.668338 + 0.743858i \(0.732994\pi\)
\(798\) 4.00000i 0.141598i
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 16.0000i 0.564980i
\(803\) 28.0000i 0.988099i
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 3.00000 0.105670
\(807\) 18.0000i 0.633630i
\(808\) 0 0
\(809\) −20.0000 −0.703163 −0.351581 0.936157i \(-0.614356\pi\)
−0.351581 + 0.936157i \(0.614356\pi\)
\(810\) 0 0
\(811\) −14.0000 −0.491606 −0.245803 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) − 1.00000i − 0.0350931i
\(813\) 20.0000i 0.701431i
\(814\) 12.0000 0.420600
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) − 4.00000i − 0.139942i
\(818\) 14.0000i 0.489499i
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) 4.00000i 0.139516i
\(823\) 18.0000i 0.627441i 0.949515 + 0.313720i \(0.101575\pi\)
−0.949515 + 0.313720i \(0.898425\pi\)
\(824\) 17.0000 0.592223
\(825\) 0 0
\(826\) 3.00000 0.104383
\(827\) − 50.0000i − 1.73867i −0.494223 0.869335i \(-0.664547\pi\)
0.494223 0.869335i \(-0.335453\pi\)
\(828\) − 7.00000i − 0.243267i
\(829\) −3.00000 −0.104194 −0.0520972 0.998642i \(-0.516591\pi\)
−0.0520972 + 0.998642i \(0.516591\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 1.00000i 0.0346688i
\(833\) 1.00000i 0.0346479i
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 8.00000 0.276686
\(837\) 3.00000i 0.103695i
\(838\) 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
\(841\) −28.0000 −0.965517
\(842\) 34.0000i 1.17172i
\(843\) 0 0
\(844\) −15.0000 −0.516321
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 7.00000i 0.240523i
\(848\) − 11.0000i − 0.377742i
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) −42.0000 −1.43974
\(852\) 4.00000i 0.137038i
\(853\) − 43.0000i − 1.47229i −0.676823 0.736146i \(-0.736644\pi\)
0.676823 0.736146i \(-0.263356\pi\)
\(854\) 5.00000 0.171096
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) − 46.0000i − 1.57133i −0.618652 0.785665i \(-0.712321\pi\)
0.618652 0.785665i \(-0.287679\pi\)
\(858\) − 2.00000i − 0.0682789i
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 3.00000 0.102240
\(862\) 27.0000i 0.919624i
\(863\) − 32.0000i − 1.08929i −0.838666 0.544646i \(-0.816664\pi\)
0.838666 0.544646i \(-0.183336\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) − 16.0000i − 0.543388i
\(868\) 3.00000i 0.101827i
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) − 4.00000i − 0.135457i
\(873\) 10.0000i 0.338449i
\(874\) −28.0000 −0.947114
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) 32.0000i 1.08056i 0.841484 + 0.540282i \(0.181682\pi\)
−0.841484 + 0.540282i \(0.818318\pi\)
\(878\) − 7.00000i − 0.236239i
\(879\) 0 0
\(880\) 0 0
\(881\) 5.00000 0.168454 0.0842271 0.996447i \(-0.473158\pi\)
0.0842271 + 0.996447i \(0.473158\pi\)
\(882\) 1.00000i 0.0336718i
\(883\) − 29.0000i − 0.975928i −0.872864 0.487964i \(-0.837740\pi\)
0.872864 0.487964i \(-0.162260\pi\)
\(884\) 1.00000 0.0336336
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 50.0000i 1.67884i 0.543487 + 0.839418i \(0.317104\pi\)
−0.543487 + 0.839418i \(0.682896\pi\)
\(888\) 6.00000i 0.201347i
\(889\) 14.0000 0.469545
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) − 13.0000i − 0.435272i
\(893\) 48.0000i 1.60626i
\(894\) −5.00000 −0.167225
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 7.00000i 0.233723i
\(898\) 16.0000i 0.533927i
\(899\) −3.00000 −0.100056
\(900\) 0 0
\(901\) −11.0000 −0.366463
\(902\) − 6.00000i − 0.199778i
\(903\) − 1.00000i − 0.0332779i
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 22.0000 0.730901
\(907\) − 11.0000i − 0.365249i −0.983183 0.182625i \(-0.941541\pi\)
0.983183 0.182625i \(-0.0584593\pi\)
\(908\) − 7.00000i − 0.232303i
\(909\) 0 0
\(910\) 0 0
\(911\) −19.0000 −0.629498 −0.314749 0.949175i \(-0.601920\pi\)
−0.314749 + 0.949175i \(0.601920\pi\)
\(912\) 4.00000i 0.132453i
\(913\) 6.00000i 0.198571i
\(914\) 17.0000 0.562310
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) − 8.00000i − 0.264183i
\(918\) 1.00000i 0.0330049i
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) −22.0000 −0.724925
\(922\) 32.0000i 1.05386i
\(923\) − 4.00000i − 0.131662i
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) 36.0000 1.18303
\(927\) − 17.0000i − 0.558353i
\(928\) − 1.00000i − 0.0328266i
\(929\) −25.0000 −0.820223 −0.410112 0.912035i \(-0.634510\pi\)
−0.410112 + 0.912035i \(0.634510\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 20.0000i 0.655122i
\(933\) − 34.0000i − 1.11311i
\(934\) 39.0000 1.27612
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) − 14.0000i − 0.457360i −0.973502 0.228680i \(-0.926559\pi\)
0.973502 0.228680i \(-0.0734410\pi\)
\(938\) − 12.0000i − 0.391814i
\(939\) −18.0000 −0.587408
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 18.0000i 0.586472i
\(943\) 21.0000i 0.683854i
\(944\) 3.00000 0.0976417
\(945\) 0 0
\(946\) −2.00000 −0.0650256
\(947\) − 32.0000i − 1.03986i −0.854209 0.519930i \(-0.825958\pi\)
0.854209 0.519930i \(-0.174042\pi\)
\(948\) 2.00000i 0.0649570i
\(949\) 14.0000 0.454459
\(950\) 0 0
\(951\) −17.0000 −0.551263
\(952\) 1.00000i 0.0324102i
\(953\) 40.0000i 1.29573i 0.761756 + 0.647864i \(0.224337\pi\)
−0.761756 + 0.647864i \(0.775663\pi\)
\(954\) −11.0000 −0.356138
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 2.00000i 0.0646508i
\(958\) − 8.00000i − 0.258468i
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) − 6.00000i − 0.193448i
\(963\) 18.0000i 0.580042i
\(964\) 26.0000 0.837404
\(965\) 0 0
\(966\) −7.00000 −0.225221
\(967\) − 44.0000i − 1.41494i −0.706741 0.707472i \(-0.749835\pi\)
0.706741 0.707472i \(-0.250165\pi\)
\(968\) 7.00000i 0.224989i
\(969\) 4.00000 0.128499
\(970\) 0 0
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 4.00000i − 0.128234i
\(974\) −22.0000 −0.704925
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) 19.0000i 0.607553i
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 30.0000i 0.957338i
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 3.00000 0.0956365
\(985\) 0 0
\(986\) −1.00000 −0.0318465
\(987\) 12.0000i 0.381964i
\(988\) − 4.00000i − 0.127257i
\(989\) 7.00000 0.222587
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 3.00000i 0.0952501i
\(993\) 3.00000i 0.0952021i
\(994\) 4.00000 0.126872
\(995\) 0 0
\(996\) −3.00000 −0.0950586
\(997\) − 2.00000i − 0.0633406i −0.999498 0.0316703i \(-0.989917\pi\)
0.999498 0.0316703i \(-0.0100827\pi\)
\(998\) 27.0000i 0.854670i
\(999\) 6.00000 0.189832
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.j.799.2 2
3.2 odd 2 3150.2.g.g.2899.1 2
5.2 odd 4 1050.2.a.e.1.1 1
5.3 odd 4 1050.2.a.o.1.1 yes 1
5.4 even 2 inner 1050.2.g.j.799.1 2
15.2 even 4 3150.2.a.bl.1.1 1
15.8 even 4 3150.2.a.c.1.1 1
15.14 odd 2 3150.2.g.g.2899.2 2
20.3 even 4 8400.2.a.t.1.1 1
20.7 even 4 8400.2.a.bt.1.1 1
35.13 even 4 7350.2.a.ca.1.1 1
35.27 even 4 7350.2.a.bj.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.e.1.1 1 5.2 odd 4
1050.2.a.o.1.1 yes 1 5.3 odd 4
1050.2.g.j.799.1 2 5.4 even 2 inner
1050.2.g.j.799.2 2 1.1 even 1 trivial
3150.2.a.c.1.1 1 15.8 even 4
3150.2.a.bl.1.1 1 15.2 even 4
3150.2.g.g.2899.1 2 3.2 odd 2
3150.2.g.g.2899.2 2 15.14 odd 2
7350.2.a.bj.1.1 1 35.27 even 4
7350.2.a.ca.1.1 1 35.13 even 4
8400.2.a.t.1.1 1 20.3 even 4
8400.2.a.bt.1.1 1 20.7 even 4