Properties

Label 3150.2.g.g.2899.2
Level 3150
Weight 2
Character 3150.2899
Analytic conductor 25.153
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1050)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.2
Root \(1.00000i\)
Character \(\chi\) = 3150.2899
Dual form 3150.2.g.g.2899.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} -2.00000 q^{11} +1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} -4.00000 q^{19} -2.00000i q^{22} -7.00000i q^{23} -1.00000 q^{26} -1.00000i q^{28} +1.00000 q^{29} +3.00000 q^{31} +1.00000i q^{32} +1.00000 q^{34} +6.00000i q^{37} -4.00000i q^{38} +3.00000 q^{41} -1.00000i q^{43} +2.00000 q^{44} +7.00000 q^{46} -12.0000i q^{47} -1.00000 q^{49} -1.00000i q^{52} -11.0000i q^{53} +1.00000 q^{56} +1.00000i q^{58} -3.00000 q^{59} +5.00000 q^{61} +3.00000i q^{62} -1.00000 q^{64} +12.0000i q^{67} +1.00000i q^{68} -4.00000 q^{71} -14.0000i q^{73} -6.00000 q^{74} +4.00000 q^{76} -2.00000i q^{77} +2.00000 q^{79} +3.00000i q^{82} +3.00000i q^{83} +1.00000 q^{86} +2.00000i q^{88} +10.0000 q^{89} -1.00000 q^{91} +7.00000i q^{92} +12.0000 q^{94} +10.0000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 4q^{11} - 2q^{14} + 2q^{16} - 8q^{19} - 2q^{26} + 2q^{29} + 6q^{31} + 2q^{34} + 6q^{41} + 4q^{44} + 14q^{46} - 2q^{49} + 2q^{56} - 6q^{59} + 10q^{61} - 2q^{64} - 8q^{71} - 12q^{74} + 8q^{76} + 4q^{79} + 2q^{86} + 20q^{89} - 2q^{91} + 24q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) −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 \(-0.0386960\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 2.00000i − 0.426401i
\(23\) − 7.00000i − 1.45960i −0.683660 0.729800i \(-0.739613\pi\)
0.683660 0.729800i \(-0.260387\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) − 1.00000i − 0.188982i
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 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
\(33\) 0 0
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) − 1.00000i − 0.152499i −0.997089 0.0762493i \(-0.975706\pi\)
0.997089 0.0762493i \(-0.0242945\pi\)
\(44\) 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 \(-0.339264\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) − 1.00000i − 0.138675i
\(53\) − 11.0000i − 1.51097i −0.655168 0.755483i \(-0.727402\pi\)
0.655168 0.755483i \(-0.272598\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 1.00000i 0.131306i
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 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
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 1.00000i 0.121268i
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) − 14.0000i − 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) − 2.00000i − 0.227921i
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.00000i 0.331295i
\(83\) 3.00000i 0.329293i 0.986353 + 0.164646i \(0.0526483\pi\)
−0.986353 + 0.164646i \(0.947352\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.00000 0.107833
\(87\) 0 0
\(88\) 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
\(92\) 7.00000i 0.729800i
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) − 17.0000i − 1.67506i −0.546392 0.837530i \(-0.683999\pi\)
0.546392 0.837530i \(-0.316001\pi\)
\(104\) 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 \(-0.335922\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 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
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) − 3.00000i − 0.276172i
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 5.00000i 0.452679i
\(123\) 0 0
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) 0 0
\(127\) − 14.0000i − 1.24230i −0.783692 0.621150i \(-0.786666\pi\)
0.783692 0.621150i \(-0.213334\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) − 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 \(0.945342\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 4.00000i − 0.335673i
\(143\) − 2.00000i − 0.167248i
\(144\) 0 0
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) − 6.00000i − 0.493197i
\(149\) 5.00000 0.409616 0.204808 0.978802i \(-0.434343\pi\)
0.204808 + 0.978802i \(0.434343\pi\)
\(150\) 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
\(153\) 0 0
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 0 0
\(157\) − 18.0000i − 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 2.00000i 0.159111i
\(159\) 0 0
\(160\) 0 0
\(161\) 7.00000 0.551677
\(162\) 0 0
\(163\) − 19.0000i − 1.48819i −0.668071 0.744097i \(-0.732880\pi\)
0.668071 0.744097i \(-0.267120\pi\)
\(164\) −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 \(-0.0246562\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 1.00000i 0.0762493i
\(173\) 12.0000i 0.912343i 0.889892 + 0.456172i \(0.150780\pi\)
−0.889892 + 0.456172i \(0.849220\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 10.0000i 0.749532i
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) − 1.00000i − 0.0741249i
\(183\) 0 0
\(184\) −7.00000 −0.516047
\(185\) 0 0
\(186\) 0 0
\(187\) 2.00000i 0.146254i
\(188\) 12.0000i 0.875190i
\(189\) 0 0
\(190\) 0 0
\(191\) 13.0000 0.940647 0.470323 0.882494i \(-0.344137\pi\)
0.470323 + 0.882494i \(0.344137\pi\)
\(192\) 0 0
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) −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 \(0.588224\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.00000i 0.0701862i
\(204\) 0 0
\(205\) 0 0
\(206\) 17.0000 1.18445
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 18.0000 1.23045
\(215\) 0 0
\(216\) 0 0
\(217\) 3.00000i 0.203653i
\(218\) 4.00000i 0.270914i
\(219\) 0 0
\(220\) 0 0
\(221\) 1.00000 0.0672673
\(222\) 0 0
\(223\) − 13.0000i − 0.870544i −0.900299 0.435272i \(-0.856652\pi\)
0.900299 0.435272i \(-0.143348\pi\)
\(224\) −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 \(0.925374\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 1.00000i − 0.0656532i
\(233\) − 20.0000i − 1.31024i −0.755523 0.655122i \(-0.772617\pi\)
0.755523 0.655122i \(-0.227383\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 3.00000 0.195283
\(237\) 0 0
\(238\) 1.00000i 0.0648204i
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 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
\(243\) 0 0
\(244\) −5.00000 −0.320092
\(245\) 0 0
\(246\) 0 0
\(247\) − 4.00000i − 0.254514i
\(248\) − 3.00000i − 0.190500i
\(249\) 0 0
\(250\) 0 0
\(251\) 15.0000 0.946792 0.473396 0.880850i \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) 0 0
\(253\) 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 \(-0.0498425\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 0 0
\(262\) − 8.00000i − 0.494242i
\(263\) − 11.0000i − 0.678289i −0.940734 0.339145i \(-0.889862\pi\)
0.940734 0.339145i \(-0.110138\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) − 12.0000i − 0.733017i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 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
\(273\) 0 0
\(274\) −4.00000 −0.241649
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) 4.00000i 0.239904i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) − 20.0000i − 1.18888i −0.804141 0.594438i \(-0.797374\pi\)
0.804141 0.594438i \(-0.202626\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 3.00000i 0.177084i
\(288\) 0 0
\(289\) 16.0000 0.941176
\(290\) 0 0
\(291\) 0 0
\(292\) 14.0000i 0.819288i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 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
\(306\) 0 0
\(307\) 22.0000i 1.25561i 0.778372 + 0.627803i \(0.216046\pi\)
−0.778372 + 0.627803i \(0.783954\pi\)
\(308\) 2.00000i 0.113961i
\(309\) 0 0
\(310\) 0 0
\(311\) −34.0000 −1.92796 −0.963982 0.265969i \(-0.914308\pi\)
−0.963982 + 0.265969i \(0.914308\pi\)
\(312\) 0 0
\(313\) 18.0000i 1.01742i 0.860938 + 0.508710i \(0.169877\pi\)
−0.860938 + 0.508710i \(0.830123\pi\)
\(314\) 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 \(-0.158423\pi\)
\(318\) 0 0
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) 0 0
\(322\) 7.00000i 0.390095i
\(323\) 4.00000i 0.222566i
\(324\) 0 0
\(325\) 0 0
\(326\) 19.0000 1.05231
\(327\) 0 0
\(328\) − 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
\(333\) 0 0
\(334\) 2.00000 0.109435
\(335\) 0 0
\(336\) 0 0
\(337\) − 27.0000i − 1.47078i −0.677642 0.735392i \(-0.736998\pi\)
0.677642 0.735392i \(-0.263002\pi\)
\(338\) 12.0000i 0.652714i
\(339\) 0 0
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) − 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 \(0.754128\pi\)
\(348\) 0 0
\(349\) 1.00000 0.0535288 0.0267644 0.999642i \(-0.491480\pi\)
0.0267644 + 0.999642i \(0.491480\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 2.00000i − 0.106600i
\(353\) − 10.0000i − 0.532246i −0.963939 0.266123i \(-0.914257\pi\)
0.963939 0.266123i \(-0.0857428\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) 10.0000i 0.528516i
\(359\) −9.00000 −0.475002 −0.237501 0.971387i \(-0.576328\pi\)
−0.237501 + 0.971387i \(0.576328\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 18.0000i − 0.946059i
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) 0 0
\(366\) 0 0
\(367\) − 13.0000i − 0.678594i −0.940679 0.339297i \(-0.889811\pi\)
0.940679 0.339297i \(-0.110189\pi\)
\(368\) − 7.00000i − 0.364900i
\(369\) 0 0
\(370\) 0 0
\(371\) 11.0000 0.571092
\(372\) 0 0
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 1.00000i 0.0515026i
\(378\) 0 0
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 13.0000i 0.665138i
\(383\) 20.0000i 1.02195i 0.859595 + 0.510976i \(0.170716\pi\)
−0.859595 + 0.510976i \(0.829284\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) − 10.0000i − 0.507673i
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) −7.00000 −0.354005
\(392\) 1.00000i 0.0505076i
\(393\) 0 0
\(394\) −27.0000 −1.36024
\(395\) 0 0
\(396\) 0 0
\(397\) − 3.00000i − 0.150566i −0.997162 0.0752828i \(-0.976014\pi\)
0.997162 0.0752828i \(-0.0239860\pi\)
\(398\) 24.0000i 1.20301i
\(399\) 0 0
\(400\) 0 0
\(401\) −16.0000 −0.799002 −0.399501 0.916733i \(-0.630817\pi\)
−0.399501 + 0.916733i \(0.630817\pi\)
\(402\) 0 0
\(403\) 3.00000i 0.149441i
\(404\) 0 0
\(405\) 0 0
\(406\) −1.00000 −0.0496292
\(407\) − 12.0000i − 0.594818i
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 17.0000i 0.837530i
\(413\) − 3.00000i − 0.147620i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 8.00000i 0.391293i
\(419\) −25.0000 −1.22133 −0.610665 0.791889i \(-0.709098\pi\)
−0.610665 + 0.791889i \(0.709098\pi\)
\(420\) 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
\(423\) 0 0
\(424\) −11.0000 −0.534207
\(425\) 0 0
\(426\) 0 0
\(427\) 5.00000i 0.241967i
\(428\) 18.0000i 0.870063i
\(429\) 0 0
\(430\) 0 0
\(431\) −27.0000 −1.30054 −0.650272 0.759701i \(-0.725345\pi\)
−0.650272 + 0.759701i \(0.725345\pi\)
\(432\) 0 0
\(433\) 34.0000i 1.63394i 0.576683 + 0.816968i \(0.304347\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) −3.00000 −0.144005
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 28.0000i 1.33942i
\(438\) 0 0
\(439\) −7.00000 −0.334092 −0.167046 0.985949i \(-0.553423\pi\)
−0.167046 + 0.985949i \(0.553423\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.00000i 0.0475651i
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 13.0000 0.615568
\(447\) 0 0
\(448\) − 1.00000i − 0.0472456i
\(449\) −16.0000 −0.755087 −0.377543 0.925992i \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 6.00000i 0.282216i
\(453\) 0 0
\(454\) −7.00000 −0.328526
\(455\) 0 0
\(456\) 0 0
\(457\) 17.0000i 0.795226i 0.917553 + 0.397613i \(0.130161\pi\)
−0.917553 + 0.397613i \(0.869839\pi\)
\(458\) − 14.0000i − 0.654177i
\(459\) 0 0
\(460\) 0 0
\(461\) −32.0000 −1.49039 −0.745194 0.666847i \(-0.767643\pi\)
−0.745194 + 0.666847i \(0.767643\pi\)
\(462\) 0 0
\(463\) 36.0000i 1.67306i 0.547920 + 0.836531i \(0.315420\pi\)
−0.547920 + 0.836531i \(0.684580\pi\)
\(464\) 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 \(-0.358161\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) 3.00000i 0.138086i
\(473\) 2.00000i 0.0919601i
\(474\) 0 0
\(475\) 0 0
\(476\) −1.00000 −0.0458349
\(477\) 0 0
\(478\) − 24.0000i − 1.09773i
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) − 26.0000i − 1.18427i
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 0 0
\(487\) − 22.0000i − 0.996915i −0.866914 0.498458i \(-0.833900\pi\)
0.866914 0.498458i \(-0.166100\pi\)
\(488\) − 5.00000i − 0.226339i
\(489\) 0 0
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 0 0
\(493\) − 1.00000i − 0.0450377i
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) 3.00000 0.134704
\(497\) − 4.00000i − 0.179425i
\(498\) 0 0
\(499\) 27.0000 1.20869 0.604343 0.796724i \(-0.293436\pi\)
0.604343 + 0.796724i \(0.293436\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 15.0000i 0.669483i
\(503\) 2.00000i 0.0891756i 0.999005 + 0.0445878i \(0.0141974\pi\)
−0.999005 + 0.0445878i \(0.985803\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −14.0000 −0.622376
\(507\) 0 0
\(508\) 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
\(513\) 0 0
\(514\) 5.00000 0.220541
\(515\) 0 0
\(516\) 0 0
\(517\) 24.0000i 1.05552i
\(518\) − 6.00000i − 0.263625i
\(519\) 0 0
\(520\) 0 0
\(521\) −39.0000 −1.70862 −0.854311 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) 0 0
\(523\) − 8.00000i − 0.349816i −0.984585 0.174908i \(-0.944037\pi\)
0.984585 0.174908i \(-0.0559627\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) 11.0000 0.479623
\(527\) − 3.00000i − 0.130682i
\(528\) 0 0
\(529\) −26.0000 −1.13043
\(530\) 0 0
\(531\) 0 0
\(532\) 4.00000i 0.173422i
\(533\) 3.00000i 0.129944i
\(534\) 0 0
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) 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
\(543\) 0 0
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) 0 0
\(547\) 15.0000i 0.641354i 0.947189 + 0.320677i \(0.103910\pi\)
−0.947189 + 0.320677i \(0.896090\pi\)
\(548\) − 4.00000i − 0.170872i
\(549\) 0 0
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) 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 \(0.875462\pi\)
\(558\) 0 0
\(559\) 1.00000 0.0422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 41.0000i − 1.72794i −0.503540 0.863972i \(-0.667969\pi\)
0.503540 0.863972i \(-0.332031\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) 4.00000i 0.167836i
\(569\) 14.0000 0.586911 0.293455 0.955973i \(-0.405195\pi\)
0.293455 + 0.955973i \(0.405195\pi\)
\(570\) 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
\(573\) 0 0
\(574\) −3.00000 −0.125218
\(575\) 0 0
\(576\) 0 0
\(577\) − 34.0000i − 1.41544i −0.706494 0.707719i \(-0.749724\pi\)
0.706494 0.707719i \(-0.250276\pi\)
\(578\) 16.0000i 0.665512i
\(579\) 0 0
\(580\) 0 0
\(581\) −3.00000 −0.124461
\(582\) 0 0
\(583\) 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 \(-0.188126\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 0 0
\(592\) 6.00000i 0.246598i
\(593\) 18.0000i 0.739171i 0.929197 + 0.369586i \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −5.00000 −0.204808
\(597\) 0 0
\(598\) 7.00000i 0.286251i
\(599\) −45.0000 −1.83865 −0.919325 0.393499i \(-0.871265\pi\)
−0.919325 + 0.393499i \(0.871265\pi\)
\(600\) 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
\(603\) 0 0
\(604\) −22.0000 −0.895167
\(605\) 0 0
\(606\) 0 0
\(607\) − 28.0000i − 1.13648i −0.822861 0.568242i \(-0.807624\pi\)
0.822861 0.568242i \(-0.192376\pi\)
\(608\) − 4.00000i − 0.162221i
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) 18.0000i 0.727013i 0.931592 + 0.363507i \(0.118421\pi\)
−0.931592 + 0.363507i \(0.881579\pi\)
\(614\) −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 \(0.909065\pi\)
\(618\) 0 0
\(619\) −38.0000 −1.52735 −0.763674 0.645601i \(-0.776607\pi\)
−0.763674 + 0.645601i \(0.776607\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 34.0000i − 1.36328i
\(623\) 10.0000i 0.400642i
\(624\) 0 0
\(625\) 0 0
\(626\) −18.0000 −0.719425
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 17.0000 0.675156
\(635\) 0 0
\(636\) 0 0
\(637\) − 1.00000i − 0.0396214i
\(638\) − 2.00000i − 0.0791808i
\(639\) 0 0
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) 2.00000i 0.0788723i 0.999222 + 0.0394362i \(0.0125562\pi\)
−0.999222 + 0.0394362i \(0.987444\pi\)
\(644\) −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 \(0.962370\pi\)
\(648\) 0 0
\(649\) 6.00000 0.235521
\(650\) 0 0
\(651\) 0 0
\(652\) 19.0000i 0.744097i
\(653\) − 14.0000i − 0.547862i −0.961749 0.273931i \(-0.911676\pi\)
0.961749 0.273931i \(-0.0883240\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.00000 0.117130
\(657\) 0 0
\(658\) 12.0000i 0.467809i
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 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
\(663\) 0 0
\(664\) 3.00000 0.116423
\(665\) 0 0
\(666\) 0 0
\(667\) − 7.00000i − 0.271041i
\(668\) 2.00000i 0.0773823i
\(669\) 0 0
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) 27.0000i 1.04077i 0.853931 + 0.520387i \(0.174212\pi\)
−0.853931 + 0.520387i \(0.825788\pi\)
\(674\) 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 \(-0.125568\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 0 0
\(682\) − 6.00000i − 0.229752i
\(683\) 6.00000i 0.229584i 0.993390 + 0.114792i \(0.0366201\pi\)
−0.993390 + 0.114792i \(0.963380\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) − 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
\(693\) 0 0
\(694\) −26.0000 −0.986947
\(695\) 0 0
\(696\) 0 0
\(697\) − 3.00000i − 0.113633i
\(698\) 1.00000i 0.0378506i
\(699\) 0 0
\(700\) 0 0
\(701\) 39.0000 1.47301 0.736505 0.676432i \(-0.236475\pi\)
0.736505 + 0.676432i \(0.236475\pi\)
\(702\) 0 0
\(703\) − 24.0000i − 0.905177i
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) 0 0
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 10.0000i − 0.374766i
\(713\) − 21.0000i − 0.786456i
\(714\) 0 0
\(715\) 0 0
\(716\) −10.0000 −0.373718
\(717\) 0 0
\(718\) − 9.00000i − 0.335877i
\(719\) −34.0000 −1.26799 −0.633993 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) 0 0
\(721\) 17.0000 0.633113
\(722\) − 3.00000i − 0.111648i
\(723\) 0 0
\(724\) 18.0000 0.668965
\(725\) 0 0
\(726\) 0 0
\(727\) 1.00000i 0.0370879i 0.999828 + 0.0185440i \(0.00590307\pi\)
−0.999828 + 0.0185440i \(0.994097\pi\)
\(728\) 1.00000i 0.0370625i
\(729\) 0 0
\(730\) 0 0
\(731\) −1.00000 −0.0369863
\(732\) 0 0
\(733\) 19.0000i 0.701781i 0.936416 + 0.350891i \(0.114121\pi\)
−0.936416 + 0.350891i \(0.885879\pi\)
\(734\) 13.0000 0.479839
\(735\) 0 0
\(736\) 7.00000 0.258023
\(737\) − 24.0000i − 0.884051i
\(738\) 0 0
\(739\) −11.0000 −0.404642 −0.202321 0.979319i \(-0.564848\pi\)
−0.202321 + 0.979319i \(0.564848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 11.0000i 0.403823i
\(743\) 3.00000i 0.110059i 0.998485 + 0.0550297i \(0.0175253\pi\)
−0.998485 + 0.0550297i \(0.982475\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) 0 0
\(748\) − 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
\(753\) 0 0
\(754\) −1.00000 −0.0364179
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 5.00000i 0.181608i
\(759\) 0 0
\(760\) 0 0
\(761\) −34.0000 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(762\) 0 0
\(763\) 4.00000i 0.144810i
\(764\) −13.0000 −0.470323
\(765\) 0 0
\(766\) −20.0000 −0.722629
\(767\) − 3.00000i − 0.108324i
\(768\) 0 0
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 2.00000i − 0.0719816i
\(773\) 20.0000i 0.719350i 0.933078 + 0.359675i \(0.117112\pi\)
−0.933078 + 0.359675i \(0.882888\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) 22.0000i 0.788738i
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) − 7.00000i − 0.250319i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) − 38.0000i − 1.35455i −0.735728 0.677277i \(-0.763160\pi\)
0.735728 0.677277i \(-0.236840\pi\)
\(788\) − 27.0000i − 0.961835i
\(789\) 0 0
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 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 \(0.732994\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) 0 0
\(802\) − 16.0000i − 0.564980i
\(803\) 28.0000i 0.988099i
\(804\) 0 0
\(805\) 0 0
\(806\) −3.00000 −0.105670
\(807\) 0 0
\(808\) 0 0
\(809\) 20.0000 0.703163 0.351581 0.936157i \(-0.385644\pi\)
0.351581 + 0.936157i \(0.385644\pi\)
\(810\) 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
\(813\) 0 0
\(814\) 12.0000 0.420600
\(815\) 0 0
\(816\) 0 0
\(817\) 4.00000i 0.139942i
\(818\) 14.0000i 0.489499i
\(819\) 0 0
\(820\) 0 0
\(821\) 38.0000 1.32621 0.663105 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) 0 0
\(823\) − 18.0000i − 0.627441i −0.949515 0.313720i \(-0.898425\pi\)
0.949515 0.313720i \(-0.101575\pi\)
\(824\) −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 \(-0.335453\pi\)
\(828\) 0 0
\(829\) −3.00000 −0.104194 −0.0520972 0.998642i \(-0.516591\pi\)
−0.0520972 + 0.998642i \(0.516591\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 1.00000i − 0.0346688i
\(833\) 1.00000i 0.0346479i
\(834\) 0 0
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) 0 0
\(838\) − 25.0000i − 0.863611i
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 34.0000i 1.17172i
\(843\) 0 0
\(844\) −15.0000 −0.516321
\(845\) 0 0
\(846\) 0 0
\(847\) − 7.00000i − 0.240523i
\(848\) − 11.0000i − 0.377742i
\(849\) 0 0
\(850\) 0 0
\(851\) 42.0000 1.43974
\(852\) 0 0
\(853\) 43.0000i 1.47229i 0.676823 + 0.736146i \(0.263356\pi\)
−0.676823 + 0.736146i \(0.736644\pi\)
\(854\) −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 \(-0.287679\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 27.0000i − 0.919624i
\(863\) − 32.0000i − 1.08929i −0.838666 0.544646i \(-0.816664\pi\)
0.838666 0.544646i \(-0.183336\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) − 3.00000i − 0.101827i
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) − 4.00000i − 0.135457i
\(873\) 0 0
\(874\) −28.0000 −0.947114
\(875\) 0 0
\(876\) 0 0
\(877\) − 32.0000i − 1.08056i −0.841484 0.540282i \(-0.818318\pi\)
0.841484 0.540282i \(-0.181682\pi\)
\(878\) − 7.00000i − 0.236239i
\(879\) 0 0
\(880\) 0 0
\(881\) −5.00000 −0.168454 −0.0842271 0.996447i \(-0.526842\pi\)
−0.0842271 + 0.996447i \(0.526842\pi\)
\(882\) 0 0
\(883\) 29.0000i 0.975928i 0.872864 + 0.487964i \(0.162260\pi\)
−0.872864 + 0.487964i \(0.837740\pi\)
\(884\) −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 \(0.682896\pi\)
\(888\) 0 0
\(889\) 14.0000 0.469545
\(890\) 0 0
\(891\) 0 0
\(892\) 13.0000i 0.435272i
\(893\) 48.0000i 1.60626i
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) − 16.0000i − 0.533927i
\(899\) 3.00000 0.100056
\(900\) 0 0
\(901\) −11.0000 −0.366463
\(902\) − 6.00000i − 0.199778i
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) 11.0000i 0.365249i 0.983183 + 0.182625i \(0.0584593\pi\)
−0.983183 + 0.182625i \(0.941541\pi\)
\(908\) − 7.00000i − 0.232303i
\(909\) 0 0
\(910\) 0 0
\(911\) 19.0000 0.629498 0.314749 0.949175i \(-0.398080\pi\)
0.314749 + 0.949175i \(0.398080\pi\)
\(912\) 0 0
\(913\) − 6.00000i − 0.198571i
\(914\) −17.0000 −0.562310
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) − 8.00000i − 0.264183i
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 32.0000i − 1.05386i
\(923\) − 4.00000i − 0.131662i
\(924\) 0 0
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) 0 0
\(928\) 1.00000i 0.0328266i
\(929\) 25.0000 0.820223 0.410112 0.912035i \(-0.365490\pi\)
0.410112 + 0.912035i \(0.365490\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 20.0000i 0.655122i
\(933\) 0 0
\(934\) 39.0000 1.27612
\(935\) 0 0
\(936\) 0 0
\(937\) 14.0000i 0.457360i 0.973502 + 0.228680i \(0.0734410\pi\)
−0.973502 + 0.228680i \(0.926559\pi\)
\(938\) − 12.0000i − 0.391814i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) − 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 \(-0.174042\pi\)
\(948\) 0 0
\(949\) 14.0000 0.454459
\(950\) 0 0
\(951\) 0 0
\(952\) − 1.00000i − 0.0324102i
\(953\) 40.0000i 1.29573i 0.761756 + 0.647864i \(0.224337\pi\)
−0.761756 + 0.647864i \(0.775663\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 8.00000i 0.258468i
\(959\) −4.00000 −0.129167
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) − 6.00000i − 0.193448i
\(963\) 0 0
\(964\) 26.0000 0.837404
\(965\) 0 0
\(966\) 0 0
\(967\) 44.0000i 1.41494i 0.706741 + 0.707472i \(0.250165\pi\)
−0.706741 + 0.707472i \(0.749835\pi\)
\(968\) 7.00000i 0.224989i
\(969\) 0 0
\(970\) 0 0
\(971\) 44.0000 1.41203 0.706014 0.708198i \(-0.250492\pi\)
0.706014 + 0.708198i \(0.250492\pi\)
\(972\) 0 0
\(973\) 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 \(0.765500\pi\)
\(978\) 0 0
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) 0 0
\(982\) − 30.0000i − 0.957338i
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 1.00000 0.0318465
\(987\) 0 0
\(988\) 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
\(993\) 0 0
\(994\) 4.00000 0.126872
\(995\) 0 0
\(996\) 0 0
\(997\) 2.00000i 0.0633406i 0.999498 + 0.0316703i \(0.0100827\pi\)
−0.999498 + 0.0316703i \(0.989917\pi\)
\(998\) 27.0000i 0.854670i
\(999\) 0 0
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 3150.2.g.g.2899.2 2
3.2 odd 2 1050.2.g.j.799.1 2
5.2 odd 4 3150.2.a.c.1.1 1
5.3 odd 4 3150.2.a.bl.1.1 1
5.4 even 2 inner 3150.2.g.g.2899.1 2
15.2 even 4 1050.2.a.o.1.1 yes 1
15.8 even 4 1050.2.a.e.1.1 1
15.14 odd 2 1050.2.g.j.799.2 2
60.23 odd 4 8400.2.a.bt.1.1 1
60.47 odd 4 8400.2.a.t.1.1 1
105.62 odd 4 7350.2.a.ca.1.1 1
105.83 odd 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 15.8 even 4
1050.2.a.o.1.1 yes 1 15.2 even 4
1050.2.g.j.799.1 2 3.2 odd 2
1050.2.g.j.799.2 2 15.14 odd 2
3150.2.a.c.1.1 1 5.2 odd 4
3150.2.a.bl.1.1 1 5.3 odd 4
3150.2.g.g.2899.1 2 5.4 even 2 inner
3150.2.g.g.2899.2 2 1.1 even 1 trivial
7350.2.a.bj.1.1 1 105.83 odd 4
7350.2.a.ca.1.1 1 105.62 odd 4
8400.2.a.t.1.1 1 60.47 odd 4
8400.2.a.bt.1.1 1 60.23 odd 4