Properties

Label 3450.2.d.u.2899.1
Level $3450$
Weight $2$
Character 3450.2899
Analytic conductor $27.548$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3450.2899
Dual form 3450.2.d.u.2899.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +6.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} +4.00000 q^{19} +2.00000 q^{21} -6.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} +2.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} +6.00000i q^{33} +1.00000 q^{36} -4.00000i q^{37} -4.00000i q^{38} -2.00000 q^{39} +2.00000 q^{41} -2.00000i q^{42} +8.00000i q^{43} -6.00000 q^{44} +1.00000 q^{46} +1.00000i q^{48} +3.00000 q^{49} -2.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} +4.00000i q^{57} -2.00000i q^{58} +4.00000 q^{59} +8.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +6.00000 q^{66} -1.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} -6.00000i q^{73} -4.00000 q^{74} -4.00000 q^{76} -12.0000i q^{77} +2.00000i q^{78} +14.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +6.00000i q^{83} -2.00000 q^{84} +8.00000 q^{86} +2.00000i q^{87} +6.00000i q^{88} +16.0000 q^{89} +4.00000 q^{91} -1.00000i q^{92} -8.00000i q^{93} +1.00000 q^{96} +2.00000i q^{97} -3.00000i q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + 12q^{11} - 4q^{14} + 2q^{16} + 8q^{19} + 4q^{21} - 2q^{24} + 4q^{26} + 4q^{29} - 16q^{31} + 2q^{36} - 4q^{39} + 4q^{41} - 12q^{44} + 2q^{46} + 6q^{49} - 2q^{54} + 4q^{56} + 8q^{59} - 2q^{64} + 12q^{66} - 2q^{69} - 16q^{71} - 8q^{74} - 8q^{76} + 28q^{79} + 2q^{81} - 4q^{84} + 16q^{86} + 32q^{89} + 8q^{91} + 2q^{96} - 12q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) − 6.00000i − 1.27920i
\(23\) 1.00000i 0.208514i
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) − 1.00000i − 0.192450i
\(28\) 2.00000i 0.377964i
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 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
\(33\) 6.00000i 1.04447i
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 4.00000i − 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 4.00000i 0.529813i
\(58\) − 2.00000i − 0.262613i
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 8.00000i 1.01600i
\(63\) 2.00000i 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) − 12.0000i − 1.36753i
\(78\) 2.00000i 0.226455i
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 2.00000i − 0.220863i
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 2.00000i 0.214423i
\(88\) 6.00000i 0.639602i
\(89\) 16.0000 1.69600 0.847998 0.529999i \(-0.177808\pi\)
0.847998 + 0.529999i \(0.177808\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) − 1.00000i − 0.104257i
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 10.0000i 0.985329i 0.870219 + 0.492665i \(0.163977\pi\)
−0.870219 + 0.492665i \(0.836023\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) − 2.00000i − 0.188982i
\(113\) − 16.0000i − 1.50515i −0.658505 0.752577i \(-0.728811\pi\)
0.658505 0.752577i \(-0.271189\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) − 2.00000i − 0.184900i
\(118\) − 4.00000i − 0.368230i
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 2.00000i 0.180334i
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) − 12.0000i − 1.06483i −0.846484 0.532414i \(-0.821285\pi\)
0.846484 0.532414i \(-0.178715\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) − 6.00000i − 0.522233i
\(133\) − 8.00000i − 0.693688i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 20.0000i 1.70872i 0.519685 + 0.854358i \(0.326049\pi\)
−0.519685 + 0.854358i \(0.673951\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000i 0.671345i
\(143\) 12.0000i 1.00349i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −6.00000 −0.496564
\(147\) 3.00000i 0.247436i
\(148\) 4.00000i 0.328798i
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 4.00000i 0.324443i
\(153\) 0 0
\(154\) −12.0000 −0.966988
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) − 4.00000i − 0.319235i −0.987179 0.159617i \(-0.948974\pi\)
0.987179 0.159617i \(-0.0510260\pi\)
\(158\) − 14.0000i − 1.11378i
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) − 1.00000i − 0.0785674i
\(163\) − 12.0000i − 0.939913i −0.882690 0.469956i \(-0.844270\pi\)
0.882690 0.469956i \(-0.155730\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 2.00000i 0.154303i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) − 8.00000i − 0.609994i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 6.00000 0.452267
\(177\) 4.00000i 0.300658i
\(178\) − 16.0000i − 1.19925i
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 4.00000 0.297318 0.148659 0.988889i \(-0.452504\pi\)
0.148659 + 0.988889i \(0.452504\pi\)
\(182\) − 4.00000i − 0.296500i
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 18.0000i 1.29567i 0.761781 + 0.647834i \(0.224325\pi\)
−0.761781 + 0.647834i \(0.775675\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) − 22.0000i − 1.56744i −0.621117 0.783718i \(-0.713321\pi\)
0.621117 0.783718i \(-0.286679\pi\)
\(198\) 6.00000i 0.426401i
\(199\) 22.0000 1.55954 0.779769 0.626067i \(-0.215336\pi\)
0.779769 + 0.626067i \(0.215336\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 2.00000i − 0.140720i
\(203\) − 4.00000i − 0.280745i
\(204\) 0 0
\(205\) 0 0
\(206\) 10.0000 0.696733
\(207\) − 1.00000i − 0.0695048i
\(208\) 2.00000i 0.138675i
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) 28.0000 1.92760 0.963800 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) − 2.00000i − 0.137361i
\(213\) − 8.00000i − 0.548151i
\(214\) −6.00000 −0.410152
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 16.0000i 1.08615i
\(218\) 12.0000i 0.812743i
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 0 0
\(222\) − 4.00000i − 0.268462i
\(223\) 28.0000i 1.87502i 0.347960 + 0.937509i \(0.386874\pi\)
−0.347960 + 0.937509i \(0.613126\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −16.0000 −1.06430
\(227\) 10.0000i 0.663723i 0.943328 + 0.331862i \(0.107677\pi\)
−0.943328 + 0.331862i \(0.892323\pi\)
\(228\) − 4.00000i − 0.264906i
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) 2.00000i 0.131306i
\(233\) − 2.00000i − 0.131024i −0.997852 0.0655122i \(-0.979132\pi\)
0.997852 0.0655122i \(-0.0208681\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 14.0000i 0.909398i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) − 25.0000i − 1.60706i
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 8.00000i 0.509028i
\(248\) − 8.00000i − 0.508001i
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) 6.00000i 0.377217i
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 6.00000i − 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) 8.00000i 0.498058i
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) − 8.00000i − 0.494242i
\(263\) − 4.00000i − 0.246651i −0.992366 0.123325i \(-0.960644\pi\)
0.992366 0.123325i \(-0.0393559\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) −8.00000 −0.490511
\(267\) 16.0000i 0.979184i
\(268\) 0 0
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) 4.00000i 0.242091i
\(274\) 20.0000 1.20824
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 10.0000i 0.600842i 0.953807 + 0.300421i \(0.0971271\pi\)
−0.953807 + 0.300421i \(0.902873\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) − 28.0000i − 1.66443i −0.554455 0.832214i \(-0.687073\pi\)
0.554455 0.832214i \(-0.312927\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) − 4.00000i − 0.236113i
\(288\) 1.00000i 0.0589256i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 6.00000i 0.351123i
\(293\) − 22.0000i − 1.28525i −0.766179 0.642627i \(-0.777845\pi\)
0.766179 0.642627i \(-0.222155\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) − 6.00000i − 0.348155i
\(298\) 10.0000i 0.579284i
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) − 20.0000i − 1.15087i
\(303\) 2.00000i 0.114897i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 12.0000i 0.683763i
\(309\) −10.0000 −0.568880
\(310\) 0 0
\(311\) −32.0000 −1.81455 −0.907277 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) − 2.00000i − 0.113228i
\(313\) 18.0000i 1.01742i 0.860938 + 0.508710i \(0.169877\pi\)
−0.860938 + 0.508710i \(0.830123\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) −14.0000 −0.787562
\(317\) 30.0000i 1.68497i 0.538721 + 0.842484i \(0.318908\pi\)
−0.538721 + 0.842484i \(0.681092\pi\)
\(318\) 2.00000i 0.112154i
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) − 2.00000i − 0.111456i
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) − 12.0000i − 0.663602i
\(328\) 2.00000i 0.110432i
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) − 6.00000i − 0.329293i
\(333\) 4.00000i 0.219199i
\(334\) 0 0
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 22.0000i 1.19842i 0.800593 + 0.599208i \(0.204518\pi\)
−0.800593 + 0.599208i \(0.795482\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) 16.0000 0.869001
\(340\) 0 0
\(341\) −48.0000 −2.59935
\(342\) 4.00000i 0.216295i
\(343\) − 20.0000i − 1.07990i
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) − 32.0000i − 1.71785i −0.512101 0.858925i \(-0.671133\pi\)
0.512101 0.858925i \(-0.328867\pi\)
\(348\) − 2.00000i − 0.107211i
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) − 6.00000i − 0.319801i
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −16.0000 −0.847998
\(357\) 0 0
\(358\) 0 0
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 4.00000i − 0.210235i
\(363\) 25.0000i 1.31216i
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) 2.00000i 0.104399i 0.998637 + 0.0521996i \(0.0166232\pi\)
−0.998637 + 0.0521996i \(0.983377\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 8.00000i 0.414781i
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.00000i 0.206010i
\(378\) 2.00000i 0.102869i
\(379\) 36.0000 1.84920 0.924598 0.380945i \(-0.124401\pi\)
0.924598 + 0.380945i \(0.124401\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) − 8.00000i − 0.409316i
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) − 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
\(391\) 0 0
\(392\) 3.00000i 0.151523i
\(393\) 8.00000i 0.403547i
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) 14.0000i 0.702640i 0.936255 + 0.351320i \(0.114267\pi\)
−0.936255 + 0.351320i \(0.885733\pi\)
\(398\) − 22.0000i − 1.10276i
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) 8.00000 0.399501 0.199750 0.979847i \(-0.435987\pi\)
0.199750 + 0.979847i \(0.435987\pi\)
\(402\) 0 0
\(403\) − 16.0000i − 0.797017i
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) − 24.0000i − 1.18964i
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −20.0000 −0.986527
\(412\) − 10.0000i − 0.492665i
\(413\) − 8.00000i − 0.393654i
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 4.00000i 0.195881i
\(418\) − 24.0000i − 1.17388i
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) 24.0000 1.16969 0.584844 0.811146i \(-0.301156\pi\)
0.584844 + 0.811146i \(0.301156\pi\)
\(422\) − 28.0000i − 1.36302i
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) 6.00000i 0.290021i
\(429\) −12.0000 −0.579365
\(430\) 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
\(433\) 26.0000i 1.24948i 0.780833 + 0.624740i \(0.214795\pi\)
−0.780833 + 0.624740i \(0.785205\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) 4.00000i 0.191346i
\(438\) − 6.00000i − 0.286691i
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) 28.0000 1.32584
\(447\) − 10.0000i − 0.472984i
\(448\) 2.00000i 0.0944911i
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 16.0000i 0.752577i
\(453\) 20.0000i 0.939682i
\(454\) 10.0000 0.469323
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 18.0000i 0.842004i 0.907060 + 0.421002i \(0.138322\pi\)
−0.907060 + 0.421002i \(0.861678\pi\)
\(458\) 4.00000i 0.186908i
\(459\) 0 0
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) − 12.0000i − 0.558291i
\(463\) − 32.0000i − 1.48717i −0.668644 0.743583i \(-0.733125\pi\)
0.668644 0.743583i \(-0.266875\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −2.00000 −0.0926482
\(467\) 30.0000i 1.38823i 0.719862 + 0.694117i \(0.244205\pi\)
−0.719862 + 0.694117i \(0.755795\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) 0 0
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 4.00000i 0.184115i
\(473\) 48.0000i 2.20704i
\(474\) 14.0000 0.643041
\(475\) 0 0
\(476\) 0 0
\(477\) − 2.00000i − 0.0915737i
\(478\) 0 0
\(479\) 40.0000 1.82765 0.913823 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) − 18.0000i − 0.819878i
\(483\) 2.00000i 0.0910032i
\(484\) −25.0000 −1.13636
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 32.0000i − 1.45006i −0.688718 0.725029i \(-0.741826\pi\)
0.688718 0.725029i \(-0.258174\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) − 2.00000i − 0.0901670i
\(493\) 0 0
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 16.0000i 0.717698i
\(498\) 6.00000i 0.268866i
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 18.0000i 0.803379i
\(503\) − 28.0000i − 1.24846i −0.781241 0.624229i \(-0.785413\pi\)
0.781241 0.624229i \(-0.214587\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 6.00000 0.266733
\(507\) 9.00000i 0.399704i
\(508\) 12.0000i 0.532414i
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) − 1.00000i − 0.0441942i
\(513\) − 4.00000i − 0.176604i
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 8.00000i 0.351500i
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) − 40.0000i − 1.74908i −0.484955 0.874539i \(-0.661164\pi\)
0.484955 0.874539i \(-0.338836\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) 0 0
\(528\) 6.00000i 0.261116i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 8.00000i 0.346844i
\(533\) 4.00000i 0.173259i
\(534\) 16.0000 0.692388
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) − 30.0000i − 1.29339i
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −6.00000 −0.257960 −0.128980 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(542\) 4.00000i 0.171815i
\(543\) 4.00000i 0.171656i
\(544\) 0 0
\(545\) 0 0
\(546\) 4.00000 0.171184
\(547\) − 12.0000i − 0.513083i −0.966533 0.256541i \(-0.917417\pi\)
0.966533 0.256541i \(-0.0825830\pi\)
\(548\) − 20.0000i − 0.854358i
\(549\) 0 0
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) − 1.00000i − 0.0425628i
\(553\) − 28.0000i − 1.19068i
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 10.0000i 0.423714i 0.977301 + 0.211857i \(0.0679510\pi\)
−0.977301 + 0.211857i \(0.932049\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 16.0000i 0.674919i
\(563\) − 18.0000i − 0.758610i −0.925272 0.379305i \(-0.876163\pi\)
0.925272 0.379305i \(-0.123837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) − 2.00000i − 0.0839921i
\(568\) − 8.00000i − 0.335673i
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) − 12.0000i − 0.501745i
\(573\) 8.00000i 0.334205i
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 14.0000i − 0.582828i −0.956597 0.291414i \(-0.905874\pi\)
0.956597 0.291414i \(-0.0941257\pi\)
\(578\) − 17.0000i − 0.707107i
\(579\) −18.0000 −0.748054
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 2.00000i 0.0829027i
\(583\) 12.0000i 0.496989i
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −22.0000 −0.908812
\(587\) 24.0000i 0.990586i 0.868726 + 0.495293i \(0.164939\pi\)
−0.868726 + 0.495293i \(0.835061\pi\)
\(588\) − 3.00000i − 0.123718i
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) − 4.00000i − 0.164399i
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 22.0000i 0.900400i
\(598\) 2.00000i 0.0817861i
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) − 16.0000i − 0.652111i
\(603\) 0 0
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) 2.00000 0.0812444
\(607\) 4.00000i 0.162355i 0.996700 + 0.0811775i \(0.0258681\pi\)
−0.996700 + 0.0811775i \(0.974132\pi\)
\(608\) − 4.00000i − 0.162221i
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 16.0000i 0.646234i 0.946359 + 0.323117i \(0.104731\pi\)
−0.946359 + 0.323117i \(0.895269\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 12.0000 0.483494
\(617\) − 40.0000i − 1.61034i −0.593045 0.805170i \(-0.702074\pi\)
0.593045 0.805170i \(-0.297926\pi\)
\(618\) 10.0000i 0.402259i
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 32.0000i 1.28308i
\(623\) − 32.0000i − 1.28205i
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 18.0000 0.719425
\(627\) 24.0000i 0.958468i
\(628\) 4.00000i 0.159617i
\(629\) 0 0
\(630\) 0 0
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 14.0000i 0.556890i
\(633\) 28.0000i 1.11290i
\(634\) 30.0000 1.19145
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) 6.00000i 0.237729i
\(638\) − 12.0000i − 0.475085i
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) − 6.00000i − 0.236801i
\(643\) 16.0000i 0.630978i 0.948929 + 0.315489i \(0.102169\pi\)
−0.948929 + 0.315489i \(0.897831\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) 12.0000i 0.469956i
\(653\) − 34.0000i − 1.33052i −0.746611 0.665261i \(-0.768320\pi\)
0.746611 0.665261i \(-0.231680\pi\)
\(654\) −12.0000 −0.469237
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) 40.0000 1.55582 0.777910 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) 20.0000i 0.777322i
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 2.00000i 0.0774403i
\(668\) 0 0
\(669\) −28.0000 −1.08254
\(670\) 0 0
\(671\) 0 0
\(672\) − 2.00000i − 0.0771517i
\(673\) 30.0000i 1.15642i 0.815890 + 0.578208i \(0.196248\pi\)
−0.815890 + 0.578208i \(0.803752\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) − 16.0000i − 0.614476i
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) −10.0000 −0.383201
\(682\) 48.0000i 1.83801i
\(683\) − 16.0000i − 0.612223i −0.951996 0.306111i \(-0.900972\pi\)
0.951996 0.306111i \(-0.0990280\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) − 4.00000i − 0.152610i
\(688\) 8.00000i 0.304997i
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 6.00000i 0.228086i
\(693\) 12.0000i 0.455842i
\(694\) −32.0000 −1.21470
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) 0 0
\(698\) 10.0000i 0.378506i
\(699\) 2.00000 0.0756469
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) − 2.00000i − 0.0754851i
\(703\) − 16.0000i − 0.603451i
\(704\) −6.00000 −0.226134
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) − 4.00000i − 0.150435i
\(708\) − 4.00000i − 0.150329i
\(709\) −24.0000 −0.901339 −0.450669 0.892691i \(-0.648815\pi\)
−0.450669 + 0.892691i \(0.648815\pi\)
\(710\) 0 0
\(711\) −14.0000 −0.525041
\(712\) 16.0000i 0.599625i
\(713\) − 8.00000i − 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 32.0000i 1.19423i
\(719\) −32.0000 −1.19340 −0.596699 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) 0 0
\(721\) 20.0000 0.744839
\(722\) 3.00000i 0.111648i
\(723\) 18.0000i 0.669427i
\(724\) −4.00000 −0.148659
\(725\) 0 0
\(726\) 25.0000 0.927837
\(727\) − 2.00000i − 0.0741759i −0.999312 0.0370879i \(-0.988192\pi\)
0.999312 0.0370879i \(-0.0118082\pi\)
\(728\) 4.00000i 0.148250i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 48.0000i 1.77292i 0.462805 + 0.886460i \(0.346843\pi\)
−0.462805 + 0.886460i \(0.653157\pi\)
\(734\) 2.00000 0.0738213
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 2.00000i 0.0736210i
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) − 4.00000i − 0.146845i
\(743\) − 52.0000i − 1.90769i −0.300291 0.953847i \(-0.597084\pi\)
0.300291 0.953847i \(-0.402916\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) − 6.00000i − 0.219529i
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) 0 0
\(753\) − 18.0000i − 0.655956i
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 20.0000i 0.726912i 0.931611 + 0.363456i \(0.118403\pi\)
−0.931611 + 0.363456i \(0.881597\pi\)
\(758\) − 36.0000i − 1.30758i
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) − 12.0000i − 0.434714i
\(763\) 24.0000i 0.868858i
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) 8.00000i 0.288863i
\(768\) 1.00000i 0.0360844i
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) − 18.0000i − 0.647834i
\(773\) 42.0000i 1.51064i 0.655359 + 0.755318i \(0.272517\pi\)
−0.655359 + 0.755318i \(0.727483\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) − 8.00000i − 0.286998i
\(778\) 10.0000i 0.358517i
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) −48.0000 −1.71758
\(782\) 0 0
\(783\) − 2.00000i − 0.0714742i
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 8.00000 0.285351
\(787\) 8.00000i 0.285169i 0.989783 + 0.142585i \(0.0455413\pi\)
−0.989783 + 0.142585i \(0.954459\pi\)
\(788\) 22.0000i 0.783718i
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) −32.0000 −1.13779
\(792\) − 6.00000i − 0.213201i
\(793\) 0 0
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) −22.0000 −0.779769
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) − 8.00000i − 0.283197i
\(799\) 0 0
\(800\) 0 0
\(801\) −16.0000 −0.565332
\(802\) − 8.00000i − 0.282490i
\(803\) − 36.0000i − 1.27041i
\(804\) 0 0
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 30.0000i 1.05605i
\(808\) 2.00000i 0.0703598i
\(809\) 50.0000 1.75791 0.878953 0.476908i \(-0.158243\pi\)
0.878953 + 0.476908i \(0.158243\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) 4.00000i 0.140372i
\(813\) − 4.00000i − 0.140286i
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) 0 0
\(817\) 32.0000i 1.11954i
\(818\) − 10.0000i − 0.349642i
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 20.0000i 0.697580i
\(823\) 32.0000i 1.11545i 0.830026 + 0.557725i \(0.188326\pi\)
−0.830026 + 0.557725i \(0.811674\pi\)
\(824\) −10.0000 −0.348367
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 30.0000i 1.04320i 0.853189 + 0.521601i \(0.174665\pi\)
−0.853189 + 0.521601i \(0.825335\pi\)
\(828\) 1.00000i 0.0347524i
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) − 2.00000i − 0.0693375i
\(833\) 0 0
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) −24.0000 −0.830057
\(837\) 8.00000i 0.276520i
\(838\) 26.0000i 0.898155i
\(839\) 44.0000 1.51905 0.759524 0.650479i \(-0.225432\pi\)
0.759524 + 0.650479i \(0.225432\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) − 24.0000i − 0.827095i
\(843\) − 16.0000i − 0.551069i
\(844\) −28.0000 −0.963800
\(845\) 0 0
\(846\) 0 0
\(847\) − 50.0000i − 1.71802i
\(848\) 2.00000i 0.0686803i
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) 8.00000i 0.274075i
\(853\) 18.0000i 0.616308i 0.951336 + 0.308154i \(0.0997113\pi\)
−0.951336 + 0.308154i \(0.900289\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) 10.0000i 0.341593i 0.985306 + 0.170797i \(0.0546341\pi\)
−0.985306 + 0.170797i \(0.945366\pi\)
\(858\) 12.0000i 0.409673i
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) 4.00000 0.136320
\(862\) 36.0000i 1.22616i
\(863\) 48.0000i 1.63394i 0.576681 + 0.816970i \(0.304348\pi\)
−0.576681 + 0.816970i \(0.695652\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) 17.0000i 0.577350i
\(868\) − 16.0000i − 0.543075i
\(869\) 84.0000 2.84950
\(870\) 0 0
\(871\) 0 0
\(872\) − 12.0000i − 0.406371i
\(873\) − 2.00000i − 0.0676897i
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) − 6.00000i − 0.202606i −0.994856 0.101303i \(-0.967699\pi\)
0.994856 0.101303i \(-0.0323011\pi\)
\(878\) 0 0
\(879\) 22.0000 0.742042
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 20.0000i 0.673054i 0.941674 + 0.336527i \(0.109252\pi\)
−0.941674 + 0.336527i \(0.890748\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) − 40.0000i − 1.34307i −0.740973 0.671534i \(-0.765636\pi\)
0.740973 0.671534i \(-0.234364\pi\)
\(888\) 4.00000i 0.134231i
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) 6.00000 0.201008
\(892\) − 28.0000i − 0.937509i
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) − 2.00000i − 0.0667781i
\(898\) 10.0000i 0.333704i
\(899\) −16.0000 −0.533630
\(900\) 0 0
\(901\) 0 0
\(902\) − 12.0000i − 0.399556i
\(903\) 16.0000i 0.532447i
\(904\) 16.0000 0.532152
\(905\) 0 0
\(906\) 20.0000 0.664455
\(907\) − 44.0000i − 1.46100i −0.682915 0.730498i \(-0.739288\pi\)
0.682915 0.730498i \(-0.260712\pi\)
\(908\) − 10.0000i − 0.331862i
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 4.00000i 0.132453i
\(913\) 36.0000i 1.19143i
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) − 16.0000i − 0.528367i
\(918\) 0 0
\(919\) 30.0000 0.989609 0.494804 0.869004i \(-0.335240\pi\)
0.494804 + 0.869004i \(0.335240\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 14.0000i 0.461065i
\(923\) − 16.0000i − 0.526646i
\(924\) −12.0000 −0.394771
\(925\) 0 0
\(926\) −32.0000 −1.05159
\(927\) − 10.0000i − 0.328443i
\(928\) − 2.00000i − 0.0656532i
\(929\) −2.00000 −0.0656179 −0.0328089 0.999462i \(-0.510445\pi\)
−0.0328089 + 0.999462i \(0.510445\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) 2.00000i 0.0655122i
\(933\) − 32.0000i − 1.04763i
\(934\) 30.0000 0.981630
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 2.00000i 0.0653372i 0.999466 + 0.0326686i \(0.0104006\pi\)
−0.999466 + 0.0326686i \(0.989599\pi\)
\(938\) 0 0
\(939\) −18.0000 −0.587408
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) − 4.00000i − 0.130327i
\(943\) 2.00000i 0.0651290i
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) 16.0000i 0.519930i 0.965618 + 0.259965i \(0.0837111\pi\)
−0.965618 + 0.259965i \(0.916289\pi\)
\(948\) − 14.0000i − 0.454699i
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) −30.0000 −0.972817
\(952\) 0 0
\(953\) − 24.0000i − 0.777436i −0.921357 0.388718i \(-0.872918\pi\)
0.921357 0.388718i \(-0.127082\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) 0 0
\(957\) 12.0000i 0.387905i
\(958\) − 40.0000i − 1.29234i
\(959\) 40.0000 1.29167
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) − 8.00000i − 0.257930i
\(963\) 6.00000i 0.193347i
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) 2.00000 0.0643489
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) 25.0000i 0.803530i
\(969\) 0 0
\(970\) 0 0
\(971\) 50.0000 1.60458 0.802288 0.596937i \(-0.203616\pi\)
0.802288 + 0.596937i \(0.203616\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 8.00000i − 0.256468i
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) 0 0
\(977\) 28.0000i 0.895799i 0.894084 + 0.447900i \(0.147828\pi\)
−0.894084 + 0.447900i \(0.852172\pi\)
\(978\) − 12.0000i − 0.383718i
\(979\) 96.0000 3.06817
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) 32.0000i 1.02116i
\(983\) − 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
\(986\) 0 0
\(987\) 0 0
\(988\) − 8.00000i − 0.254514i
\(989\) −8.00000 −0.254385
\(990\) 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
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) 34.0000i 1.07679i 0.842692 + 0.538395i \(0.180969\pi\)
−0.842692 + 0.538395i \(0.819031\pi\)
\(998\) 36.0000i 1.13956i
\(999\) −4.00000 −0.126554
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 3450.2.d.u.2899.1 2
5.2 odd 4 3450.2.a.ba.1.1 1
5.3 odd 4 690.2.a.a.1.1 1
5.4 even 2 inner 3450.2.d.u.2899.2 2
15.8 even 4 2070.2.a.q.1.1 1
20.3 even 4 5520.2.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.a.1.1 1 5.3 odd 4
2070.2.a.q.1.1 1 15.8 even 4
3450.2.a.ba.1.1 1 5.2 odd 4
3450.2.d.u.2899.1 2 1.1 even 1 trivial
3450.2.d.u.2899.2 2 5.4 even 2 inner
5520.2.a.y.1.1 1 20.3 even 4