Properties

Label 2850.2.d.i.799.1
Level $2850$
Weight $2$
Character 2850.799
Analytic conductor $22.757$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.7573645761\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 570)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2850.799
Dual form 2850.2.d.i.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} +2.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -4.00000i q^{22} +4.00000i q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -6.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -1.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} -4.00000i q^{43} -4.00000 q^{44} +4.00000 q^{46} +12.0000i q^{47} -1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} -2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -1.00000i q^{57} +6.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} -4.00000i q^{67} +2.00000i q^{68} +4.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} -6.00000i q^{73} +6.00000 q^{74} -1.00000 q^{76} -2.00000i q^{78} +4.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -12.0000i q^{83} -4.00000 q^{86} +6.00000i q^{87} +4.00000i q^{88} -10.0000 q^{89} -4.00000i q^{92} -4.00000i q^{93} +12.0000 q^{94} -1.00000 q^{96} -2.00000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} + 2 q^{19} + 2 q^{24} + 4 q^{26} - 12 q^{29} + 8 q^{31} - 4 q^{34} + 2 q^{36} + 4 q^{39} + 20 q^{41} - 8 q^{44} + 8 q^{46} + 14 q^{49} - 4 q^{51} + 2 q^{54} + 24 q^{59} - 4 q^{61} - 2 q^{64} - 8 q^{66} + 8 q^{69} + 16 q^{71} + 12 q^{74} - 2 q^{76} + 8 q^{79} + 2 q^{81} - 8 q^{86} - 20 q^{89} + 24 q^{94} - 2 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times\).

\(n\) \(1027\) \(1351\) \(1901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) − 4.00000i − 0.852803i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 4.00000i − 0.696311i
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) − 2.00000i − 0.277350i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.00000i − 0.132453i
\(58\) 6.00000i 0.787839i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) − 2.00000i − 0.226455i
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 10.0000i − 1.10432i
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 6.00000i 0.643268i
\(88\) 4.00000i 0.426401i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 4.00000i − 0.417029i
\(93\) − 4.00000i − 0.414781i
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) − 7.00000i − 0.707107i
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 2.00000i 0.198030i
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) − 2.00000i − 0.184900i
\(118\) − 12.0000i − 1.10469i
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 2.00000i 0.181071i
\(123\) − 10.0000i − 0.901670i
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 4.00000i 0.354943i 0.984126 + 0.177471i \(0.0567917\pi\)
−0.984126 + 0.177471i \(0.943208\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) − 10.0000i − 0.854358i −0.904167 0.427179i \(-0.859507\pi\)
0.904167 0.427179i \(-0.140493\pi\)
\(138\) − 4.00000i − 0.340503i
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) − 8.00000i − 0.671345i
\(143\) 8.00000i 0.668994i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −6.00000 −0.496564
\(147\) − 7.00000i − 0.577350i
\(148\) − 6.00000i − 0.493197i
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) − 4.00000i − 0.318223i
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) − 1.00000i − 0.0785674i
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 4.00000i 0.304997i
\(173\) − 10.0000i − 0.760286i −0.924928 0.380143i \(-0.875875\pi\)
0.924928 0.380143i \(-0.124125\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) − 12.0000i − 0.901975i
\(178\) 10.0000i 0.749532i
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) − 8.00000i − 0.585018i
\(188\) − 12.0000i − 0.875190i
\(189\) 0 0
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) − 2.00000i − 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) 4.00000i 0.284268i
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) − 10.0000i − 0.703598i
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) − 4.00000i − 0.278019i
\(208\) 2.00000i 0.138675i
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) − 8.00000i − 0.548151i
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) − 6.00000i − 0.406371i
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) − 6.00000i − 0.402694i
\(223\) 12.0000i 0.803579i 0.915732 + 0.401790i \(0.131612\pi\)
−0.915732 + 0.401790i \(0.868388\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 1.00000i 0.0662266i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 6.00000i − 0.393919i
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) − 4.00000i − 0.259828i
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) − 1.00000i − 0.0641500i
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) 2.00000i 0.127257i
\(248\) 4.00000i 0.254000i
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 16.0000i 1.00591i
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 2.00000i − 0.124757i −0.998053 0.0623783i \(-0.980131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) 4.00000i 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 12.0000i 0.741362i
\(263\) − 4.00000i − 0.246651i −0.992366 0.123325i \(-0.960644\pi\)
0.992366 0.123325i \(-0.0393559\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) 10.0000i 0.611990i
\(268\) 4.00000i 0.244339i
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) − 20.0000i − 1.19952i
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) − 12.0000i − 0.714590i
\(283\) − 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
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 6.00000i 0.351123i
\(293\) − 2.00000i − 0.116841i −0.998292 0.0584206i \(-0.981394\pi\)
0.998292 0.0584206i \(-0.0186065\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 4.00000i 0.232104i
\(298\) 18.0000i 1.04271i
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) 12.0000i 0.690522i
\(303\) − 10.0000i − 0.574485i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) − 4.00000i − 0.228292i −0.993464 0.114146i \(-0.963587\pi\)
0.993464 0.114146i \(-0.0364132\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 2.00000i 0.113228i
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 26.0000i 1.46031i 0.683284 + 0.730153i \(0.260551\pi\)
−0.683284 + 0.730153i \(0.739449\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) − 2.00000i − 0.111283i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) − 6.00000i − 0.331801i
\(328\) 10.0000i 0.552158i
\(329\) 0 0
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 12.0000i 0.658586i
\(333\) − 6.00000i − 0.328798i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 26.0000i − 1.41631i −0.706057 0.708155i \(-0.749528\pi\)
0.706057 0.708155i \(-0.250472\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 1.00000i 0.0540738i
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) − 6.00000i − 0.321634i
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) − 4.00000i − 0.213201i
\(353\) 26.0000i 1.38384i 0.721974 + 0.691920i \(0.243235\pi\)
−0.721974 + 0.691920i \(0.756765\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 20.0000i 1.05703i
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 18.0000i − 0.946059i
\(363\) − 5.00000i − 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) − 24.0000i − 1.25279i −0.779506 0.626395i \(-0.784530\pi\)
0.779506 0.626395i \(-0.215470\pi\)
\(368\) 4.00000i 0.208514i
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 4.00000i 0.207390i
\(373\) − 22.0000i − 1.13912i −0.821951 0.569558i \(-0.807114\pi\)
0.821951 0.569558i \(-0.192886\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) − 12.0000i − 0.618031i
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) − 20.0000i − 1.02329i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 4.00000i 0.203331i
\(388\) 2.00000i 0.101535i
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 7.00000i 0.353553i
\(393\) 12.0000i 0.605320i
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 18.0000i 0.903394i 0.892171 + 0.451697i \(0.149181\pi\)
−0.892171 + 0.451697i \(0.850819\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 4.00000i 0.199502i
\(403\) 8.00000i 0.398508i
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0000i 1.18964i
\(408\) − 2.00000i − 0.0990148i
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) 4.00000i 0.197066i
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) − 20.0000i − 0.979404i
\(418\) − 4.00000i − 0.195646i
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 4.00000i 0.194717i
\(423\) − 12.0000i − 0.583460i
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) − 4.00000i − 0.193347i
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 34.0000i 1.63394i 0.576683 + 0.816968i \(0.304347\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 4.00000i 0.191346i
\(438\) 6.00000i 0.286691i
\(439\) −12.0000 −0.572729 −0.286364 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) − 4.00000i − 0.190261i
\(443\) 4.00000i 0.190046i 0.995475 + 0.0950229i \(0.0302924\pi\)
−0.995475 + 0.0950229i \(0.969708\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) 12.0000 0.568216
\(447\) 18.0000i 0.851371i
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) 14.0000i 0.658505i
\(453\) 12.0000i 0.563809i
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 6.00000i 0.280668i 0.990104 + 0.140334i \(0.0448177\pi\)
−0.990104 + 0.140334i \(0.955182\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −38.0000 −1.76984 −0.884918 0.465746i \(-0.845786\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) 0 0
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) 28.0000i 1.29569i 0.761774 + 0.647843i \(0.224329\pi\)
−0.761774 + 0.647843i \(0.775671\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 12.0000i 0.552345i
\(473\) − 16.0000i − 0.735681i
\(474\) −4.00000 −0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) − 6.00000i − 0.274721i
\(478\) 12.0000i 0.548867i
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) − 26.0000i − 1.18427i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 20.0000i 0.906287i 0.891438 + 0.453143i \(0.149697\pi\)
−0.891438 + 0.453143i \(0.850303\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 10.0000i 0.450835i
\(493\) 12.0000i 0.540453i
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 12.0000i 0.537733i
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 20.0000i 0.892644i
\(503\) − 36.0000i − 1.60516i −0.596544 0.802580i \(-0.703460\pi\)
0.596544 0.802580i \(-0.296540\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) − 9.00000i − 0.399704i
\(508\) − 4.00000i − 0.177471i
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 1.00000i 0.0441511i
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 48.0000i 2.11104i
\(518\) 0 0
\(519\) −10.0000 −0.438951
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) − 6.00000i − 0.262613i
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) − 8.00000i − 0.348485i
\(528\) − 4.00000i − 0.174078i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 20.0000i 0.866296i
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 20.0000i 0.863064i
\(538\) − 2.00000i − 0.0862261i
\(539\) 28.0000 1.20605
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) − 8.00000i − 0.343629i
\(543\) − 18.0000i − 0.772454i
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 10.0000i 0.427179i
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 4.00000i 0.170251i
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) − 42.0000i − 1.77960i −0.456354 0.889799i \(-0.650845\pi\)
0.456354 0.889799i \(-0.349155\pi\)
\(558\) 4.00000i 0.169334i
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 6.00000i 0.253095i
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) 8.00000i 0.335673i
\(569\) 38.0000 1.59304 0.796521 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) − 8.00000i − 0.334497i
\(573\) − 20.0000i − 0.835512i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 30.0000i 1.24892i 0.781058 + 0.624458i \(0.214680\pi\)
−0.781058 + 0.624458i \(0.785320\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) 0 0
\(582\) 2.00000i 0.0829027i
\(583\) 24.0000i 0.993978i
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) 28.0000i 1.15568i 0.816149 + 0.577842i \(0.196105\pi\)
−0.816149 + 0.577842i \(0.803895\pi\)
\(588\) 7.00000i 0.288675i
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 6.00000i 0.246598i
\(593\) 10.0000i 0.410651i 0.978694 + 0.205325i \(0.0658253\pi\)
−0.978694 + 0.205325i \(0.934175\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 0 0
\(598\) 8.00000i 0.327144i
\(599\) −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 \(0.782263\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 12.0000 0.488273
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) − 4.00000i − 0.162355i −0.996700 0.0811775i \(-0.974132\pi\)
0.996700 0.0811775i \(-0.0258681\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) 0 0
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) − 2.00000i − 0.0808452i
\(613\) 22.0000i 0.888572i 0.895885 + 0.444286i \(0.146543\pi\)
−0.895885 + 0.444286i \(0.853457\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) 14.0000i 0.563619i 0.959470 + 0.281809i \(0.0909346\pi\)
−0.959470 + 0.281809i \(0.909065\pi\)
\(618\) 4.00000i 0.160904i
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 28.0000i 1.12270i
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) − 4.00000i − 0.159745i
\(628\) − 2.00000i − 0.0798087i
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 4.00000i 0.159111i
\(633\) 4.00000i 0.158986i
\(634\) 26.0000 1.03259
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 14.0000i 0.554700i
\(638\) 24.0000i 0.950169i
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) − 4.00000i − 0.157867i
\(643\) − 4.00000i − 0.157745i −0.996885 0.0788723i \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −2.00000 −0.0786889
\(647\) − 36.0000i − 1.41531i −0.706560 0.707653i \(-0.749754\pi\)
0.706560 0.707653i \(-0.250246\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) − 6.00000i − 0.234798i −0.993085 0.117399i \(-0.962544\pi\)
0.993085 0.117399i \(-0.0374557\pi\)
\(654\) −6.00000 −0.234619
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) − 28.0000i − 1.08825i
\(663\) − 4.00000i − 0.155347i
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) − 24.0000i − 0.929284i
\(668\) 0 0
\(669\) 12.0000 0.463947
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) − 14.0000i − 0.539660i −0.962908 0.269830i \(-0.913032\pi\)
0.962908 0.269830i \(-0.0869676\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 42.0000i 1.61419i 0.590421 + 0.807096i \(0.298962\pi\)
−0.590421 + 0.807096i \(0.701038\pi\)
\(678\) 14.0000i 0.537667i
\(679\) 0 0
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) − 16.0000i − 0.612672i
\(683\) 36.0000i 1.37750i 0.724998 + 0.688751i \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 0 0
\(687\) − 10.0000i − 0.381524i
\(688\) − 4.00000i − 0.152499i
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 10.0000i 0.380143i
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) − 20.0000i − 0.757554i
\(698\) − 10.0000i − 0.378506i
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 2.00000i 0.0754851i
\(703\) 6.00000i 0.226294i
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) 0 0
\(708\) 12.0000i 0.450988i
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) − 10.0000i − 0.374766i
\(713\) 16.0000i 0.599205i
\(714\) 0 0
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) 12.0000i 0.448148i
\(718\) 12.0000i 0.447836i
\(719\) −28.0000 −1.04422 −0.522112 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 1.00000i − 0.0372161i
\(723\) − 26.0000i − 0.966950i
\(724\) −18.0000 −0.668965
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) − 40.0000i − 1.48352i −0.670667 0.741759i \(-0.733992\pi\)
0.670667 0.741759i \(-0.266008\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) − 2.00000i − 0.0739221i
\(733\) 46.0000i 1.69905i 0.527549 + 0.849524i \(0.323111\pi\)
−0.527549 + 0.849524i \(0.676889\pi\)
\(734\) −24.0000 −0.885856
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) − 16.0000i − 0.589368i
\(738\) 10.0000i 0.368105i
\(739\) −52.0000 −1.91285 −0.956425 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) 16.0000i 0.586983i 0.955962 + 0.293492i \(0.0948173\pi\)
−0.955962 + 0.293492i \(0.905183\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) 12.0000i 0.439057i
\(748\) 8.00000i 0.292509i
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 12.0000i 0.437595i
\(753\) 20.0000i 0.728841i
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) − 38.0000i − 1.38113i −0.723269 0.690567i \(-0.757361\pi\)
0.723269 0.690567i \(-0.242639\pi\)
\(758\) − 20.0000i − 0.726433i
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) − 4.00000i − 0.144905i
\(763\) 0 0
\(764\) −20.0000 −0.723575
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000i 0.866590i
\(768\) − 1.00000i − 0.0360844i
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) − 2.00000i − 0.0719816i
\(773\) 30.0000i 1.07903i 0.841978 + 0.539513i \(0.181391\pi\)
−0.841978 + 0.539513i \(0.818609\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) − 22.0000i − 0.788738i
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) − 8.00000i − 0.286079i
\(783\) − 6.00000i − 0.214423i
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 52.0000i 1.85360i 0.375555 + 0.926800i \(0.377452\pi\)
−0.375555 + 0.926800i \(0.622548\pi\)
\(788\) 2.00000i 0.0712470i
\(789\) −4.00000 −0.142404
\(790\) 0 0
\(791\) 0 0
\(792\) − 4.00000i − 0.142134i
\(793\) − 4.00000i − 0.142044i
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) 0 0
\(797\) − 54.0000i − 1.91278i −0.292096 0.956389i \(-0.594353\pi\)
0.292096 0.956389i \(-0.405647\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 14.0000i 0.494357i
\(803\) − 24.0000i − 0.846942i
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) − 2.00000i − 0.0704033i
\(808\) 10.0000i 0.351799i
\(809\) −34.0000 −1.19538 −0.597688 0.801729i \(-0.703914\pi\)
−0.597688 + 0.801729i \(0.703914\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) − 8.00000i − 0.280572i
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) − 4.00000i − 0.139942i
\(818\) 18.0000i 0.629355i
\(819\) 0 0
\(820\) 0 0
\(821\) −22.0000 −0.767805 −0.383903 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) 10.0000i 0.348790i
\(823\) 8.00000i 0.278862i 0.990232 + 0.139431i \(0.0445274\pi\)
−0.990232 + 0.139431i \(0.955473\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) − 36.0000i − 1.25184i −0.779886 0.625921i \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) 4.00000i 0.139010i
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) − 2.00000i − 0.0693375i
\(833\) − 14.0000i − 0.485071i
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) 4.00000i 0.138260i
\(838\) 12.0000i 0.414533i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) − 26.0000i − 0.896019i
\(843\) 6.00000i 0.206651i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) 6.00000i 0.206041i
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 8.00000i 0.274075i
\(853\) 14.0000i 0.479351i 0.970853 + 0.239675i \(0.0770410\pi\)
−0.970853 + 0.239675i \(0.922959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) − 26.0000i − 0.888143i −0.895991 0.444072i \(-0.853534\pi\)
0.895991 0.444072i \(-0.146466\pi\)
\(858\) − 8.00000i − 0.273115i
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 8.00000i − 0.272481i
\(863\) 8.00000i 0.272323i 0.990687 + 0.136162i \(0.0434766\pi\)
−0.990687 + 0.136162i \(0.956523\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 6.00000i 0.203186i
\(873\) 2.00000i 0.0676897i
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 46.0000i 1.55331i 0.629926 + 0.776655i \(0.283085\pi\)
−0.629926 + 0.776655i \(0.716915\pi\)
\(878\) 12.0000i 0.404980i
\(879\) −2.00000 −0.0674583
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 7.00000i 0.235702i
\(883\) 20.0000i 0.673054i 0.941674 + 0.336527i \(0.109252\pi\)
−0.941674 + 0.336527i \(0.890748\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 48.0000i 1.61168i 0.592132 + 0.805841i \(0.298286\pi\)
−0.592132 + 0.805841i \(0.701714\pi\)
\(888\) 6.00000i 0.201347i
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) − 12.0000i − 0.401790i
\(893\) 12.0000i 0.401565i
\(894\) 18.0000 0.602010
\(895\) 0 0
\(896\) 0 0
\(897\) 8.00000i 0.267112i
\(898\) − 30.0000i − 1.00111i
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) − 40.0000i − 1.33185i
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 12.0000 0.398673
\(907\) − 28.0000i − 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) − 20.0000i − 0.663723i
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) − 1.00000i − 0.0331133i
\(913\) − 48.0000i − 1.58857i
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) − 2.00000i − 0.0660098i
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 38.0000i 1.25146i
\(923\) 16.0000i 0.526646i
\(924\) 0 0
\(925\) 0 0
\(926\) 8.00000 0.262896
\(927\) 4.00000i 0.131377i
\(928\) 6.00000i 0.196960i
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) − 18.0000i − 0.589610i
\(933\) 28.0000i 0.916679i
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) − 10.0000i − 0.326686i −0.986569 0.163343i \(-0.947772\pi\)
0.986569 0.163343i \(-0.0522277\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) − 2.00000i − 0.0651635i
\(943\) 40.0000i 1.30258i
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) − 52.0000i − 1.68977i −0.534946 0.844886i \(-0.679668\pi\)
0.534946 0.844886i \(-0.320332\pi\)
\(948\) 4.00000i 0.129914i
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 26.0000 0.843108
\(952\) 0 0
\(953\) 42.0000i 1.36051i 0.732974 + 0.680257i \(0.238132\pi\)
−0.732974 + 0.680257i \(0.761868\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 24.0000i 0.775810i
\(958\) 20.0000i 0.646171i
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 12.0000i 0.386896i
\(963\) − 4.00000i − 0.128898i
\(964\) −26.0000 −0.837404
\(965\) 0 0
\(966\) 0 0
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) 5.00000i 0.160706i
\(969\) −2.00000 −0.0642493
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 0 0
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) − 2.00000i − 0.0639857i −0.999488 0.0319928i \(-0.989815\pi\)
0.999488 0.0319928i \(-0.0101854\pi\)
\(978\) 4.00000i 0.127906i
\(979\) −40.0000 −1.27841
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 36.0000i 1.14881i
\(983\) 56.0000i 1.78612i 0.449935 + 0.893061i \(0.351447\pi\)
−0.449935 + 0.893061i \(0.648553\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) − 2.00000i − 0.0636285i
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) − 28.0000i − 0.888553i
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) − 38.0000i − 1.20347i −0.798695 0.601736i \(-0.794476\pi\)
0.798695 0.601736i \(-0.205524\pi\)
\(998\) − 36.0000i − 1.13956i
\(999\) −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 2850.2.d.i.799.1 2
5.2 odd 4 570.2.a.g.1.1 1
5.3 odd 4 2850.2.a.m.1.1 1
5.4 even 2 inner 2850.2.d.i.799.2 2
15.2 even 4 1710.2.a.i.1.1 1
15.8 even 4 8550.2.a.x.1.1 1
20.7 even 4 4560.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.g.1.1 1 5.2 odd 4
1710.2.a.i.1.1 1 15.2 even 4
2850.2.a.m.1.1 1 5.3 odd 4
2850.2.d.i.799.1 2 1.1 even 1 trivial
2850.2.d.i.799.2 2 5.4 even 2 inner
4560.2.a.s.1.1 1 20.7 even 4
8550.2.a.x.1.1 1 15.8 even 4