Properties

Label 2850.2.d.e.799.2
Level $2850$
Weight $2$
Character 2850.799
Analytic conductor $22.757$
Analytic rank $1$
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: \(1\)
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 570)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -6.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +8.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} +4.00000i q^{23} +1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} -2.00000 q^{29} -2.00000 q^{31} +1.00000i q^{32} -8.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} -1.00000i q^{38} +6.00000 q^{39} -12.0000 q^{41} +2.00000i q^{42} -4.00000i q^{43} -4.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -8.00000 q^{51} +6.00000i q^{52} -10.0000i q^{53} +1.00000 q^{54} -2.00000 q^{56} -1.00000i q^{57} -2.00000i q^{58} -6.00000 q^{59} -14.0000 q^{61} -2.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} -8.00000i q^{68} -4.00000 q^{69} -8.00000 q^{71} +1.00000i q^{72} +10.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +6.00000i q^{78} -14.0000 q^{79} +1.00000 q^{81} -12.0000i q^{82} -2.00000i q^{83} -2.00000 q^{84} +4.00000 q^{86} -2.00000i q^{87} -12.0000 q^{91} -4.00000i q^{92} -2.00000i q^{93} -12.0000 q^{94} -1.00000 q^{96} +2.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} + 2 q^{24} + 12 q^{26} - 4 q^{29} - 4 q^{31} - 16 q^{34} + 2 q^{36} + 12 q^{39} - 24 q^{41} - 8 q^{46} + 6 q^{49} - 16 q^{51} + 2 q^{54} - 4 q^{56} - 12 q^{59} - 28 q^{61} - 2 q^{64} - 8 q^{69} - 16 q^{71} + 4 q^{74} + 2 q^{76} - 28 q^{79} + 2 q^{81} - 4 q^{84} + 8 q^{86} - 24 q^{91} - 24 q^{94} - 2 q^{96} + O(q^{100}) \)

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
\(7\) − 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
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 8.00000i 1.94029i 0.242536 + 0.970143i \(0.422021\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 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
\(26\) 6.00000 1.17670
\(27\) − 1.00000i − 0.192450i
\(28\) 2.00000i 0.377964i
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −8.00000 −1.37199
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 2.00000i 0.308607i
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 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
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 6.00000i 0.832050i
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) − 1.00000i − 0.132453i
\(58\) − 2.00000i − 0.262613i
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) − 2.00000i − 0.254000i
\(63\) 2.00000i 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) − 8.00000i − 0.970143i
\(69\) −4.00000 −0.481543
\(70\) 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
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 6.00000i 0.679366i
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 12.0000i − 1.32518i
\(83\) − 2.00000i − 0.219529i −0.993958 0.109764i \(-0.964990\pi\)
0.993958 0.109764i \(-0.0350096\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) − 2.00000i − 0.214423i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) − 4.00000i − 0.417029i
\(93\) − 2.00000i − 0.207390i
\(94\) −12.0000 −1.23771
\(95\) 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
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) − 8.00000i − 0.792118i
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) − 16.0000i − 1.54678i −0.633932 0.773389i \(-0.718560\pi\)
0.633932 0.773389i \(-0.281440\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) − 2.00000i − 0.188982i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 6.00000i 0.554700i
\(118\) − 6.00000i − 0.552345i
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) − 14.0000i − 1.26750i
\(123\) − 12.0000i − 1.08200i
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 2.00000i 0.173422i
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 8.00000 0.685994
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) − 4.00000i − 0.340503i
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) − 8.00000i − 0.671345i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) 3.00000i 0.247436i
\(148\) 2.00000i 0.164399i
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) − 8.00000i − 0.646762i
\(154\) 0 0
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) − 4.00000i − 0.319235i −0.987179 0.159617i \(-0.948974\pi\)
0.987179 0.159617i \(-0.0510260\pi\)
\(158\) − 14.0000i − 1.11378i
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 1.00000i 0.0785674i
\(163\) − 8.00000i − 0.626608i −0.949653 0.313304i \(-0.898564\pi\)
0.949653 0.313304i \(-0.101436\pi\)
\(164\) 12.0000 0.937043
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 4.00000i 0.304997i
\(173\) − 2.00000i − 0.152057i −0.997106 0.0760286i \(-0.975776\pi\)
0.997106 0.0760286i \(-0.0242240\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 0 0
\(177\) − 6.00000i − 0.450988i
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) − 12.0000i − 0.889499i
\(183\) − 14.0000i − 1.03491i
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 2.00000 0.146647
\(187\) 0 0
\(188\) − 12.0000i − 0.875190i
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) − 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
\(196\) −3.00000 −0.214286
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) − 6.00000i − 0.422159i
\(203\) 4.00000i 0.280745i
\(204\) 8.00000 0.560112
\(205\) 0 0
\(206\) −16.0000 −1.11477
\(207\) − 4.00000i − 0.278019i
\(208\) − 6.00000i − 0.416025i
\(209\) 0 0
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 10.0000i 0.686803i
\(213\) − 8.00000i − 0.548151i
\(214\) 16.0000 1.09374
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 4.00000i 0.271538i
\(218\) 12.0000i 0.812743i
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 48.0000 3.22883
\(222\) 2.00000i 0.134231i
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 1.00000i 0.0662266i
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.00000i 0.131306i
\(233\) − 24.0000i − 1.57229i −0.618041 0.786146i \(-0.712073\pi\)
0.618041 0.786146i \(-0.287927\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) − 14.0000i − 0.909398i
\(238\) 16.0000i 1.03713i
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) − 11.0000i − 0.707107i
\(243\) 1.00000i 0.0641500i
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) 6.00000i 0.381771i
\(248\) 2.00000i 0.127000i
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 30.0000i − 1.87135i −0.352865 0.935674i \(-0.614792\pi\)
0.352865 0.935674i \(-0.385208\pi\)
\(258\) 4.00000i 0.249029i
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) 12.0000i 0.739952i 0.929041 + 0.369976i \(0.120634\pi\)
−0.929041 + 0.369976i \(0.879366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) 12.0000i 0.733017i
\(269\) 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 \(0.538768\pi\)
\(272\) 8.00000i 0.485071i
\(273\) − 12.0000i − 0.726273i
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 4.00000i 0.240337i 0.992754 + 0.120168i \(0.0383434\pi\)
−0.992754 + 0.120168i \(0.961657\pi\)
\(278\) 12.0000i 0.719712i
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) − 12.0000i − 0.714590i
\(283\) − 20.0000i − 1.18888i −0.804141 0.594438i \(-0.797374\pi\)
0.804141 0.594438i \(-0.202626\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000i 1.41668i
\(288\) − 1.00000i − 0.0589256i
\(289\) −47.0000 −2.76471
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) − 10.0000i − 0.585206i
\(293\) 18.0000i 1.05157i 0.850617 + 0.525786i \(0.176229\pi\)
−0.850617 + 0.525786i \(0.823771\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) − 10.0000i − 0.579284i
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 18.0000i 1.03578i
\(303\) − 6.00000i − 0.344691i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 8.00000 0.457330
\(307\) − 28.0000i − 1.59804i −0.601302 0.799022i \(-0.705351\pi\)
0.601302 0.799022i \(-0.294649\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) − 6.00000i − 0.339683i
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) − 2.00000i − 0.112331i −0.998421 0.0561656i \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) 10.0000i 0.560772i
\(319\) 0 0
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) 8.00000i 0.445823i
\(323\) − 8.00000i − 0.445132i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 8.00000 0.443079
\(327\) 12.0000i 0.663602i
\(328\) 12.0000i 0.662589i
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 2.00000i 0.109764i
\(333\) 2.00000i 0.109599i
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) − 2.00000i − 0.108947i −0.998515 0.0544735i \(-0.982652\pi\)
0.998515 0.0544735i \(-0.0173480\pi\)
\(338\) − 23.0000i − 1.25104i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 1.00000i 0.0540738i
\(343\) − 20.0000i − 1.07990i
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) − 10.0000i − 0.536828i −0.963304 0.268414i \(-0.913500\pi\)
0.963304 0.268414i \(-0.0864995\pi\)
\(348\) 2.00000i 0.107211i
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 0 0
\(353\) − 20.0000i − 1.06449i −0.846590 0.532246i \(-0.821348\pi\)
0.846590 0.532246i \(-0.178652\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) 0 0
\(357\) 16.0000i 0.846810i
\(358\) − 10.0000i − 0.528516i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 12.0000i − 0.630706i
\(363\) − 11.0000i − 0.577350i
\(364\) 12.0000 0.628971
\(365\) 0 0
\(366\) 14.0000 0.731792
\(367\) 34.0000i 1.77479i 0.461014 + 0.887393i \(0.347486\pi\)
−0.461014 + 0.887393i \(0.652514\pi\)
\(368\) 4.00000i 0.208514i
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) −20.0000 −1.03835
\(372\) 2.00000i 0.103695i
\(373\) 14.0000i 0.724893i 0.932005 + 0.362446i \(0.118058\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 12.0000i 0.618031i
\(378\) − 2.00000i − 0.102869i
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) 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
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) −32.0000 −1.61831
\(392\) − 3.00000i − 0.151523i
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) 8.00000i 0.401508i 0.979642 + 0.200754i \(0.0643393\pi\)
−0.979642 + 0.200754i \(0.935661\pi\)
\(398\) − 4.00000i − 0.200502i
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 12.0000i 0.598506i
\(403\) 12.0000i 0.597763i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) 0 0
\(408\) 8.00000i 0.396059i
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) − 16.0000i − 0.788263i
\(413\) 12.0000i 0.590481i
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 12.0000i 0.587643i
\(418\) 0 0
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) − 16.0000i − 0.778868i
\(423\) − 12.0000i − 0.583460i
\(424\) −10.0000 −0.485643
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 28.0000i 1.35501i
\(428\) 16.0000i 0.773389i
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 34.0000i 1.63394i 0.576683 + 0.816968i \(0.304347\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −12.0000 −0.574696
\(437\) − 4.00000i − 0.191346i
\(438\) − 10.0000i − 0.477818i
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 48.0000i 2.28313i
\(443\) 34.0000i 1.61539i 0.589601 + 0.807694i \(0.299285\pi\)
−0.589601 + 0.807694i \(0.700715\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) − 10.0000i − 0.472984i
\(448\) 2.00000i 0.0944911i
\(449\) −36.0000 −1.69895 −0.849473 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000i 0.282216i
\(453\) 18.0000i 0.845714i
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) − 6.00000i − 0.280362i
\(459\) 8.00000 0.373408
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) − 22.0000i − 1.02243i −0.859454 0.511213i \(-0.829196\pi\)
0.859454 0.511213i \(-0.170804\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) − 22.0000i − 1.01804i −0.860755 0.509019i \(-0.830008\pi\)
0.860755 0.509019i \(-0.169992\pi\)
\(468\) − 6.00000i − 0.277350i
\(469\) −24.0000 −1.10822
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 6.00000i 0.276172i
\(473\) 0 0
\(474\) 14.0000 0.643041
\(475\) 0 0
\(476\) −16.0000 −0.733359
\(477\) 10.0000i 0.457869i
\(478\) 8.00000i 0.365911i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) − 14.0000i − 0.637683i
\(483\) 8.00000i 0.364013i
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 40.0000i 1.81257i 0.422664 + 0.906287i \(0.361095\pi\)
−0.422664 + 0.906287i \(0.638905\pi\)
\(488\) 14.0000i 0.633750i
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) −16.0000 −0.722070 −0.361035 0.932552i \(-0.617576\pi\)
−0.361035 + 0.932552i \(0.617576\pi\)
\(492\) 12.0000i 0.541002i
\(493\) − 16.0000i − 0.720604i
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 16.0000i 0.717698i
\(498\) 2.00000i 0.0896221i
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) − 12.0000i − 0.535586i
\(503\) 16.0000i 0.713405i 0.934218 + 0.356702i \(0.116099\pi\)
−0.934218 + 0.356702i \(0.883901\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) − 23.0000i − 1.02147i
\(508\) 8.00000i 0.354943i
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) 1.00000i 0.0441942i
\(513\) 1.00000i 0.0441511i
\(514\) 30.0000 1.32324
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) − 4.00000i − 0.175750i
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) − 16.0000i − 0.696971i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) − 2.00000i − 0.0867110i
\(533\) 72.0000i 3.11867i
\(534\) 0 0
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) − 10.0000i − 0.431532i
\(538\) 30.0000i 1.29339i
\(539\) 0 0
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) − 4.00000i − 0.171815i
\(543\) − 12.0000i − 0.514969i
\(544\) −8.00000 −0.342997
\(545\) 0 0
\(546\) 12.0000 0.513553
\(547\) − 36.0000i − 1.53925i −0.638497 0.769624i \(-0.720443\pi\)
0.638497 0.769624i \(-0.279557\pi\)
\(548\) − 12.0000i − 0.512615i
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) 4.00000i 0.170251i
\(553\) 28.0000i 1.19068i
\(554\) −4.00000 −0.169944
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) − 2.00000i − 0.0847427i −0.999102 0.0423714i \(-0.986509\pi\)
0.999102 0.0423714i \(-0.0134913\pi\)
\(558\) 2.00000i 0.0846668i
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) − 20.0000i − 0.843649i
\(563\) − 24.0000i − 1.01148i −0.862686 0.505740i \(-0.831220\pi\)
0.862686 0.505740i \(-0.168780\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) − 2.00000i − 0.0839921i
\(568\) 8.00000i 0.335673i
\(569\) −4.00000 −0.167689 −0.0838444 0.996479i \(-0.526720\pi\)
−0.0838444 + 0.996479i \(0.526720\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 30.0000i − 1.24892i −0.781058 0.624458i \(-0.785320\pi\)
0.781058 0.624458i \(-0.214680\pi\)
\(578\) − 47.0000i − 1.95494i
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) − 2.00000i − 0.0829027i
\(583\) 0 0
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) − 26.0000i − 1.07313i −0.843857 0.536567i \(-0.819721\pi\)
0.843857 0.536567i \(-0.180279\pi\)
\(588\) − 3.00000i − 0.123718i
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) − 2.00000i − 0.0821995i
\(593\) − 36.0000i − 1.47834i −0.673517 0.739171i \(-0.735217\pi\)
0.673517 0.739171i \(-0.264783\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) − 4.00000i − 0.163709i
\(598\) 24.0000i 0.981433i
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) − 8.00000i − 0.326056i
\(603\) 12.0000i 0.488678i
\(604\) −18.0000 −0.732410
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) 24.0000i 0.974130i 0.873366 + 0.487065i \(0.161933\pi\)
−0.873366 + 0.487065i \(0.838067\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) −4.00000 −0.162088
\(610\) 0 0
\(611\) 72.0000 2.91281
\(612\) 8.00000i 0.323381i
\(613\) 16.0000i 0.646234i 0.946359 + 0.323117i \(0.104731\pi\)
−0.946359 + 0.323117i \(0.895269\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) 20.0000i 0.805170i 0.915383 + 0.402585i \(0.131888\pi\)
−0.915383 + 0.402585i \(0.868112\pi\)
\(618\) − 16.0000i − 0.643614i
\(619\) −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
\(622\) 24.0000i 0.962312i
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 4.00000i 0.159617i
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 14.0000i 0.556890i
\(633\) − 16.0000i − 0.635943i
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) − 18.0000i − 0.713186i
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −20.0000 −0.789953 −0.394976 0.918691i \(-0.629247\pi\)
−0.394976 + 0.918691i \(0.629247\pi\)
\(642\) 16.0000i 0.631470i
\(643\) 28.0000i 1.10421i 0.833774 + 0.552106i \(0.186176\pi\)
−0.833774 + 0.552106i \(0.813824\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) − 16.0000i − 0.629025i −0.949253 0.314512i \(-0.898159\pi\)
0.949253 0.314512i \(-0.101841\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 8.00000i 0.313304i
\(653\) 42.0000i 1.64359i 0.569785 + 0.821794i \(0.307026\pi\)
−0.569785 + 0.821794i \(0.692974\pi\)
\(654\) −12.0000 −0.469237
\(655\) 0 0
\(656\) −12.0000 −0.468521
\(657\) − 10.0000i − 0.390137i
\(658\) 24.0000i 0.935617i
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) 20.0000i 0.777322i
\(663\) 48.0000i 1.86417i
\(664\) −2.00000 −0.0776151
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) − 8.00000i − 0.309761i
\(668\) − 8.00000i − 0.309529i
\(669\) −4.00000 −0.154649
\(670\) 0 0
\(671\) 0 0
\(672\) 2.00000i 0.0771517i
\(673\) − 2.00000i − 0.0770943i −0.999257 0.0385472i \(-0.987727\pi\)
0.999257 0.0385472i \(-0.0122730\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) 6.00000i 0.230429i
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 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
\(686\) 20.0000 0.763604
\(687\) − 6.00000i − 0.228914i
\(688\) − 4.00000i − 0.152499i
\(689\) −60.0000 −2.28582
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 2.00000i 0.0760286i
\(693\) 0 0
\(694\) 10.0000 0.379595
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) − 96.0000i − 3.63626i
\(698\) − 14.0000i − 0.529908i
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) 14.0000 0.528773 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) − 6.00000i − 0.226455i
\(703\) 2.00000i 0.0754314i
\(704\) 0 0
\(705\) 0 0
\(706\) 20.0000 0.752710
\(707\) 12.0000i 0.451306i
\(708\) 6.00000i 0.225494i
\(709\) 18.0000 0.676004 0.338002 0.941145i \(-0.390249\pi\)
0.338002 + 0.941145i \(0.390249\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) 0 0
\(713\) − 8.00000i − 0.299602i
\(714\) −16.0000 −0.598785
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) 8.00000i 0.298765i
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) 1.00000i 0.0372161i
\(723\) − 14.0000i − 0.520666i
\(724\) 12.0000 0.445976
\(725\) 0 0
\(726\) 11.0000 0.408248
\(727\) 42.0000i 1.55769i 0.627214 + 0.778847i \(0.284195\pi\)
−0.627214 + 0.778847i \(0.715805\pi\)
\(728\) 12.0000i 0.444750i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 32.0000 1.18356
\(732\) 14.0000i 0.517455i
\(733\) − 8.00000i − 0.295487i −0.989026 0.147743i \(-0.952799\pi\)
0.989026 0.147743i \(-0.0472010\pi\)
\(734\) −34.0000 −1.25496
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 0 0
\(738\) 12.0000i 0.441726i
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) − 20.0000i − 0.734223i
\(743\) − 40.0000i − 1.46746i −0.679442 0.733729i \(-0.737778\pi\)
0.679442 0.733729i \(-0.262222\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) 2.00000i 0.0731762i
\(748\) 0 0
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) −2.00000 −0.0729810 −0.0364905 0.999334i \(-0.511618\pi\)
−0.0364905 + 0.999334i \(0.511618\pi\)
\(752\) 12.0000i 0.437595i
\(753\) − 12.0000i − 0.437304i
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 40.0000i 1.45382i 0.686730 + 0.726912i \(0.259045\pi\)
−0.686730 + 0.726912i \(0.740955\pi\)
\(758\) 16.0000i 0.581146i
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 8.00000i 0.289809i
\(763\) − 24.0000i − 0.868858i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 36.0000i 1.29988i
\(768\) 1.00000i 0.0360844i
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) − 2.00000i − 0.0719816i
\(773\) − 10.0000i − 0.359675i −0.983696 0.179838i \(-0.942443\pi\)
0.983696 0.179838i \(-0.0575572\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) − 4.00000i − 0.143499i
\(778\) − 18.0000i − 0.645331i
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) − 32.0000i − 1.14432i
\(783\) 2.00000i 0.0714742i
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) − 20.0000i − 0.712923i −0.934310 0.356462i \(-0.883983\pi\)
0.934310 0.356462i \(-0.116017\pi\)
\(788\) − 18.0000i − 0.641223i
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 84.0000i 2.98293i
\(794\) −8.00000 −0.283909
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) − 18.0000i − 0.637593i −0.947823 0.318796i \(-0.896721\pi\)
0.947823 0.318796i \(-0.103279\pi\)
\(798\) − 2.00000i − 0.0707992i
\(799\) −96.0000 −3.39624
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) −12.0000 −0.422682
\(807\) 30.0000i 1.05605i
\(808\) 6.00000i 0.211079i
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) − 4.00000i − 0.140372i
\(813\) − 4.00000i − 0.140286i
\(814\) 0 0
\(815\) 0 0
\(816\) −8.00000 −0.280056
\(817\) 4.00000i 0.139942i
\(818\) 6.00000i 0.209785i
\(819\) 12.0000 0.419314
\(820\) 0 0
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) − 12.0000i − 0.418548i
\(823\) − 14.0000i − 0.488009i −0.969774 0.244005i \(-0.921539\pi\)
0.969774 0.244005i \(-0.0784612\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 32.0000i 1.11275i 0.830932 + 0.556375i \(0.187808\pi\)
−0.830932 + 0.556375i \(0.812192\pi\)
\(828\) 4.00000i 0.139010i
\(829\) −28.0000 −0.972480 −0.486240 0.873825i \(-0.661632\pi\)
−0.486240 + 0.873825i \(0.661632\pi\)
\(830\) 0 0
\(831\) −4.00000 −0.138758
\(832\) 6.00000i 0.208013i
\(833\) 24.0000i 0.831551i
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) 0 0
\(837\) 2.00000i 0.0691301i
\(838\) 28.0000i 0.967244i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 4.00000i 0.137849i
\(843\) − 20.0000i − 0.688837i
\(844\) 16.0000 0.550743
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) 22.0000i 0.755929i
\(848\) − 10.0000i − 0.343401i
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 8.00000i 0.274075i
\(853\) 28.0000i 0.958702i 0.877623 + 0.479351i \(0.159128\pi\)
−0.877623 + 0.479351i \(0.840872\pi\)
\(854\) −28.0000 −0.958140
\(855\) 0 0
\(856\) −16.0000 −0.546869
\(857\) 42.0000i 1.43469i 0.696717 + 0.717346i \(0.254643\pi\)
−0.696717 + 0.717346i \(0.745357\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 24.0000i 0.817443i
\(863\) − 8.00000i − 0.272323i −0.990687 0.136162i \(-0.956523\pi\)
0.990687 0.136162i \(-0.0434766\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) − 47.0000i − 1.59620i
\(868\) − 4.00000i − 0.135769i
\(869\) 0 0
\(870\) 0 0
\(871\) −72.0000 −2.43963
\(872\) − 12.0000i − 0.406371i
\(873\) − 2.00000i − 0.0676897i
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) 18.0000i 0.607817i 0.952701 + 0.303908i \(0.0982917\pi\)
−0.952701 + 0.303908i \(0.901708\pi\)
\(878\) 10.0000i 0.337484i
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) − 3.00000i − 0.101015i
\(883\) 52.0000i 1.74994i 0.484178 + 0.874970i \(0.339119\pi\)
−0.484178 + 0.874970i \(0.660881\pi\)
\(884\) −48.0000 −1.61441
\(885\) 0 0
\(886\) −34.0000 −1.14225
\(887\) 32.0000i 1.07445i 0.843437 + 0.537227i \(0.180528\pi\)
−0.843437 + 0.537227i \(0.819472\pi\)
\(888\) − 2.00000i − 0.0671156i
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) − 4.00000i − 0.133930i
\(893\) − 12.0000i − 0.401565i
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 24.0000i 0.801337i
\(898\) − 36.0000i − 1.20134i
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) 80.0000 2.66519
\(902\) 0 0
\(903\) − 8.00000i − 0.266223i
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −18.0000 −0.598010
\(907\) − 36.0000i − 1.19536i −0.801735 0.597680i \(-0.796089\pi\)
0.801735 0.597680i \(-0.203911\pi\)
\(908\) − 20.0000i − 0.663723i
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) − 1.00000i − 0.0331133i
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 0 0
\(918\) 8.00000i 0.264039i
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) − 18.0000i − 0.592798i
\(923\) 48.0000i 1.57994i
\(924\) 0 0
\(925\) 0 0
\(926\) 22.0000 0.722965
\(927\) − 16.0000i − 0.525509i
\(928\) − 2.00000i − 0.0656532i
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 24.0000i 0.786146i
\(933\) 24.0000i 0.785725i
\(934\) 22.0000 0.719862
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) 2.00000i 0.0653372i 0.999466 + 0.0326686i \(0.0104006\pi\)
−0.999466 + 0.0326686i \(0.989599\pi\)
\(938\) − 24.0000i − 0.783628i
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 4.00000i 0.130327i
\(943\) − 48.0000i − 1.56310i
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) 2.00000i 0.0649913i 0.999472 + 0.0324956i \(0.0103455\pi\)
−0.999472 + 0.0324956i \(0.989654\pi\)
\(948\) 14.0000i 0.454699i
\(949\) 60.0000 1.94768
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) − 16.0000i − 0.518563i
\(953\) 6.00000i 0.194359i 0.995267 + 0.0971795i \(0.0309821\pi\)
−0.995267 + 0.0971795i \(0.969018\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) 0 0
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) − 12.0000i − 0.386896i
\(963\) 16.0000i 0.515593i
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) − 34.0000i − 1.09337i −0.837340 0.546683i \(-0.815890\pi\)
0.837340 0.546683i \(-0.184110\pi\)
\(968\) 11.0000i 0.353553i
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 24.0000i − 0.769405i
\(974\) −40.0000 −1.28168
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) − 30.0000i − 0.959785i −0.877327 0.479893i \(-0.840676\pi\)
0.877327 0.479893i \(-0.159324\pi\)
\(978\) 8.00000i 0.255812i
\(979\) 0 0
\(980\) 0 0
\(981\) −12.0000 −0.383131
\(982\) − 16.0000i − 0.510581i
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −12.0000 −0.382546
\(985\) 0 0
\(986\) 16.0000 0.509544
\(987\) 24.0000i 0.763928i
\(988\) − 6.00000i − 0.190885i
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 50.0000 1.58830 0.794151 0.607720i \(-0.207916\pi\)
0.794151 + 0.607720i \(0.207916\pi\)
\(992\) − 2.00000i − 0.0635001i
\(993\) 20.0000i 0.634681i
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) −2.00000 −0.0633724
\(997\) 44.0000i 1.39349i 0.717317 + 0.696747i \(0.245370\pi\)
−0.717317 + 0.696747i \(0.754630\pi\)
\(998\) − 28.0000i − 0.886325i
\(999\) −2.00000 −0.0632772
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2850.2.d.e.799.2 2
5.2 odd 4 2850.2.a.n.1.1 1
5.3 odd 4 570.2.a.h.1.1 1
5.4 even 2 inner 2850.2.d.e.799.1 2
15.2 even 4 8550.2.a.bh.1.1 1
15.8 even 4 1710.2.a.c.1.1 1
20.3 even 4 4560.2.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.h.1.1 1 5.3 odd 4
1710.2.a.c.1.1 1 15.8 even 4
2850.2.a.n.1.1 1 5.2 odd 4
2850.2.d.e.799.1 2 5.4 even 2 inner
2850.2.d.e.799.2 2 1.1 even 1 trivial
4560.2.a.bc.1.1 1 20.3 even 4
8550.2.a.bh.1.1 1 15.2 even 4