Properties

Label 2850.2.d.r.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: \(x^{2} + 1\)
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.r.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +6.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} -4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} -6.00000 q^{29} -6.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} +10.0000i q^{37} +1.00000i q^{38} -6.00000 q^{39} +4.00000 q^{41} -2.00000i q^{42} -12.0000i q^{43} -4.00000 q^{44} -4.00000 q^{46} +4.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -4.00000 q^{51} -6.00000i q^{52} +10.0000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -1.00000i q^{57} +6.00000i q^{58} -10.0000 q^{59} +2.00000 q^{61} +6.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} +12.0000i q^{67} -4.00000i q^{68} +4.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} +10.0000 q^{74} +1.00000 q^{76} -8.00000i q^{77} +6.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} -4.00000i q^{82} -2.00000i q^{83} -2.00000 q^{84} -12.0000 q^{86} -6.00000i q^{87} +4.00000i q^{88} +8.00000 q^{89} +12.0000 q^{91} +4.00000i q^{92} -6.00000i q^{93} +4.00000 q^{94} +1.00000 q^{96} +2.00000i q^{97} -3.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} - 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} - 2 q^{24} + 12 q^{26} - 12 q^{29} - 12 q^{31} + 8 q^{34} + 2 q^{36} - 12 q^{39} + 8 q^{41} - 8 q^{44} - 8 q^{46} + 6 q^{49} - 8 q^{51} - 2 q^{54} + 4 q^{56} - 20 q^{59} + 4 q^{61} - 2 q^{64} + 8 q^{66} + 8 q^{69} + 16 q^{71} + 20 q^{74} + 2 q^{76} - 20 q^{79} + 2 q^{81} - 4 q^{84} - 24 q^{86} + 16 q^{89} + 24 q^{91} + 8 q^{94} + 2 q^{96} - 8 q^{99} + O(q^{100}) \)

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
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) − 4.00000i − 0.852803i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) − 1.00000i − 0.192450i
\(28\) 2.00000i 0.377964i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 4.00000i 0.696311i
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 1.00000i 0.162221i
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) − 12.0000i − 1.82998i −0.403473 0.914991i \(-0.632197\pi\)
0.403473 0.914991i \(-0.367803\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) − 6.00000i − 0.832050i
\(53\) 10.0000i 1.37361i 0.726844 + 0.686803i \(0.240986\pi\)
−0.726844 + 0.686803i \(0.759014\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) − 1.00000i − 0.132453i
\(58\) 6.00000i 0.787839i
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 6.00000i 0.762001i
\(63\) 2.00000i 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) 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
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) − 8.00000i − 0.911685i
\(78\) 6.00000i 0.679366i
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 4.00000i − 0.441726i
\(83\) − 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
\(86\) −12.0000 −1.29399
\(87\) − 6.00000i − 0.643268i
\(88\) 4.00000i 0.426401i
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) 4.00000i 0.417029i
\(93\) − 6.00000i − 0.622171i
\(94\) 4.00000 0.412568
\(95\) 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
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 4.00000i 0.396059i
\(103\) 12.0000i 1.18240i 0.806527 + 0.591198i \(0.201345\pi\)
−0.806527 + 0.591198i \(0.798655\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 8.00000i 0.773389i 0.922208 + 0.386695i \(0.126383\pi\)
−0.922208 + 0.386695i \(0.873617\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) − 2.00000i − 0.188982i
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) − 6.00000i − 0.554700i
\(118\) 10.0000i 0.920575i
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) − 2.00000i − 0.181071i
\(123\) 4.00000i 0.360668i
\(124\) 6.00000 0.538816
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) − 20.0000i − 1.77471i −0.461084 0.887357i \(-0.652539\pi\)
0.461084 0.887357i \(-0.347461\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) − 4.00000i − 0.348155i
\(133\) 2.00000i 0.173422i
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) − 4.00000i − 0.340503i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) − 8.00000i − 0.671345i
\(143\) 24.0000i 2.00698i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 3.00000i 0.247436i
\(148\) − 10.0000i − 0.821995i
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) − 1.00000i − 0.0811107i
\(153\) − 4.00000i − 0.323381i
\(154\) −8.00000 −0.644658
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) 12.0000i 0.957704i 0.877896 + 0.478852i \(0.158947\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 10.0000i 0.795557i
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) − 1.00000i − 0.0785674i
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) 24.0000i 1.85718i 0.371113 + 0.928588i \(0.378976\pi\)
−0.371113 + 0.928588i \(0.621024\pi\)
\(168\) 2.00000i 0.154303i
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 12.0000i 0.914991i
\(173\) 18.0000i 1.36851i 0.729241 + 0.684257i \(0.239873\pi\)
−0.729241 + 0.684257i \(0.760127\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) − 10.0000i − 0.751646i
\(178\) − 8.00000i − 0.599625i
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) − 12.0000i − 0.889499i
\(183\) 2.00000i 0.147844i
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) 16.0000i 1.17004i
\(188\) − 4.00000i − 0.291730i
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 22.0000i 1.56744i 0.621117 + 0.783718i \(0.286679\pi\)
−0.621117 + 0.783718i \(0.713321\pi\)
\(198\) 4.00000i 0.284268i
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 10.0000i 0.703598i
\(203\) 12.0000i 0.842235i
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 12.0000 0.836080
\(207\) 4.00000i 0.278019i
\(208\) 6.00000i 0.416025i
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) − 10.0000i − 0.686803i
\(213\) 8.00000i 0.548151i
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 12.0000i 0.814613i
\(218\) − 16.0000i − 1.08366i
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) 10.0000i 0.671156i
\(223\) − 24.0000i − 1.60716i −0.595198 0.803579i \(-0.702926\pi\)
0.595198 0.803579i \(-0.297074\pi\)
\(224\) −2.00000 −0.133631
\(225\) 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
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) − 6.00000i − 0.393919i
\(233\) 12.0000i 0.786146i 0.919507 + 0.393073i \(0.128588\pi\)
−0.919507 + 0.393073i \(0.871412\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 10.0000 0.650945
\(237\) − 10.0000i − 0.649570i
\(238\) − 8.00000i − 0.518563i
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 1.00000i 0.0641500i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 4.00000 0.255031
\(247\) − 6.00000i − 0.381771i
\(248\) − 6.00000i − 0.381000i
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) − 16.0000i − 1.00591i
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) − 12.0000i − 0.747087i
\(259\) 20.0000 1.24274
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) − 12.0000i − 0.741362i
\(263\) − 12.0000i − 0.739952i −0.929041 0.369976i \(-0.879366\pi\)
0.929041 0.369976i \(-0.120634\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) 8.00000i 0.489592i
\(268\) − 12.0000i − 0.733017i
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 4.00000i 0.242536i
\(273\) 12.0000i 0.726273i
\(274\) 0 0
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) 28.0000i 1.68236i 0.540758 + 0.841178i \(0.318138\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 4.00000 0.238620 0.119310 0.992857i \(-0.461932\pi\)
0.119310 + 0.992857i \(0.461932\pi\)
\(282\) 4.00000i 0.238197i
\(283\) − 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
\(286\) 24.0000 1.41915
\(287\) − 8.00000i − 0.472225i
\(288\) 1.00000i 0.0589256i
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) − 2.00000i − 0.117041i
\(293\) − 2.00000i − 0.116841i −0.998292 0.0584206i \(-0.981394\pi\)
0.998292 0.0584206i \(-0.0186065\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) − 4.00000i − 0.232104i
\(298\) 14.0000i 0.810998i
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 2.00000i 0.115087i
\(303\) − 10.0000i − 0.574485i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 8.00000i 0.455842i
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) − 6.00000i − 0.339683i
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) 12.0000 0.677199
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 10.0000i 0.560772i
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 8.00000i 0.445823i
\(323\) − 4.00000i − 0.222566i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) 16.0000i 0.884802i
\(328\) 4.00000i 0.220863i
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 2.00000i 0.109764i
\(333\) − 10.0000i − 0.547997i
\(334\) 24.0000 1.31322
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) − 34.0000i − 1.85210i −0.377403 0.926049i \(-0.623183\pi\)
0.377403 0.926049i \(-0.376817\pi\)
\(338\) 23.0000i 1.25104i
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) −24.0000 −1.29967
\(342\) − 1.00000i − 0.0540738i
\(343\) − 20.0000i − 1.07990i
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) − 2.00000i − 0.107366i −0.998558 0.0536828i \(-0.982904\pi\)
0.998558 0.0536828i \(-0.0170960\pi\)
\(348\) 6.00000i 0.321634i
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) − 4.00000i − 0.213201i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) −10.0000 −0.531494
\(355\) 0 0
\(356\) −8.00000 −0.423999
\(357\) 8.00000i 0.423405i
\(358\) − 2.00000i − 0.105703i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) −12.0000 −0.628971
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) − 22.0000i − 1.14839i −0.818718 0.574195i \(-0.805315\pi\)
0.818718 0.574195i \(-0.194685\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) 20.0000 1.03835
\(372\) 6.00000i 0.311086i
\(373\) − 14.0000i − 0.724893i −0.932005 0.362446i \(-0.881942\pi\)
0.932005 0.362446i \(-0.118058\pi\)
\(374\) 16.0000 0.827340
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) − 36.0000i − 1.85409i
\(378\) 2.00000i 0.102869i
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 20.0000 1.02463
\(382\) 8.00000i 0.409316i
\(383\) 16.0000i 0.817562i 0.912633 + 0.408781i \(0.134046\pi\)
−0.912633 + 0.408781i \(0.865954\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 12.0000i 0.609994i
\(388\) − 2.00000i − 0.101535i
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 3.00000i 0.151523i
\(393\) 12.0000i 0.605320i
\(394\) 22.0000 1.10834
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) − 24.0000i − 1.20453i −0.798298 0.602263i \(-0.794266\pi\)
0.798298 0.602263i \(-0.205734\pi\)
\(398\) − 4.00000i − 0.200502i
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 12.0000i 0.598506i
\(403\) − 36.0000i − 1.79329i
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 40.0000i 1.98273i
\(408\) − 4.00000i − 0.198030i
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 12.0000i − 0.591198i
\(413\) 20.0000i 0.984136i
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 4.00000i 0.195881i
\(418\) 4.00000i 0.195646i
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) 8.00000i 0.389434i
\(423\) − 4.00000i − 0.194487i
\(424\) −10.0000 −0.485643
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) − 4.00000i − 0.193574i
\(428\) − 8.00000i − 0.386695i
\(429\) −24.0000 −1.15873
\(430\) 0 0
\(431\) −40.0000 −1.92673 −0.963366 0.268190i \(-0.913575\pi\)
−0.963366 + 0.268190i \(0.913575\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 30.0000i − 1.44171i −0.693087 0.720854i \(-0.743750\pi\)
0.693087 0.720854i \(-0.256250\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) 4.00000i 0.191346i
\(438\) 2.00000i 0.0955637i
\(439\) 38.0000 1.81364 0.906821 0.421517i \(-0.138502\pi\)
0.906821 + 0.421517i \(0.138502\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 24.0000i 1.14156i
\(443\) − 6.00000i − 0.285069i −0.989790 0.142534i \(-0.954475\pi\)
0.989790 0.142534i \(-0.0455251\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) −24.0000 −1.13643
\(447\) − 14.0000i − 0.662177i
\(448\) 2.00000i 0.0944911i
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 16.0000 0.753411
\(452\) − 14.0000i − 0.658505i
\(453\) − 2.00000i − 0.0939682i
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) − 22.0000i − 1.02912i −0.857455 0.514558i \(-0.827956\pi\)
0.857455 0.514558i \(-0.172044\pi\)
\(458\) − 26.0000i − 1.21490i
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) − 8.00000i − 0.372194i
\(463\) 18.0000i 0.836531i 0.908325 + 0.418265i \(0.137362\pi\)
−0.908325 + 0.418265i \(0.862638\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 12.0000 0.555889
\(467\) − 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
\(471\) −12.0000 −0.552931
\(472\) − 10.0000i − 0.460287i
\(473\) − 48.0000i − 2.20704i
\(474\) −10.0000 −0.459315
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) − 10.0000i − 0.457869i
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −60.0000 −2.73576
\(482\) − 10.0000i − 0.455488i
\(483\) − 8.00000i − 0.364013i
\(484\) −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
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) − 4.00000i − 0.180334i
\(493\) − 24.0000i − 1.08091i
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) − 16.0000i − 0.717698i
\(498\) − 2.00000i − 0.0896221i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) 0 0
\(503\) 8.00000i 0.356702i 0.983967 + 0.178351i \(0.0570763\pi\)
−0.983967 + 0.178351i \(0.942924\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) −16.0000 −0.711287
\(507\) − 23.0000i − 1.02147i
\(508\) 20.0000i 0.887357i
\(509\) −38.0000 −1.68432 −0.842160 0.539227i \(-0.818716\pi\)
−0.842160 + 0.539227i \(0.818716\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) − 1.00000i − 0.0441942i
\(513\) 1.00000i 0.0441511i
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −12.0000 −0.528271
\(517\) 16.0000i 0.703679i
\(518\) − 20.0000i − 0.878750i
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 40.0000 1.75243 0.876216 0.481919i \(-0.160060\pi\)
0.876216 + 0.481919i \(0.160060\pi\)
\(522\) − 6.00000i − 0.262613i
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) − 24.0000i − 1.04546i
\(528\) 4.00000i 0.174078i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) − 2.00000i − 0.0867110i
\(533\) 24.0000i 1.03956i
\(534\) 8.00000 0.346194
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 2.00000i 0.0863064i
\(538\) 14.0000i 0.603583i
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) − 20.0000i − 0.859074i
\(543\) 0 0
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) 12.0000 0.513553
\(547\) − 4.00000i − 0.171028i −0.996337 0.0855138i \(-0.972747\pi\)
0.996337 0.0855138i \(-0.0272532\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 4.00000i 0.170251i
\(553\) 20.0000i 0.850487i
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 10.0000i 0.423714i 0.977301 + 0.211857i \(0.0679510\pi\)
−0.977301 + 0.211857i \(0.932049\pi\)
\(558\) − 6.00000i − 0.254000i
\(559\) 72.0000 3.04528
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) − 4.00000i − 0.168730i
\(563\) − 24.0000i − 1.01148i −0.862686 0.505740i \(-0.831220\pi\)
0.862686 0.505740i \(-0.168780\pi\)
\(564\) 4.00000 0.168430
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) − 2.00000i − 0.0839921i
\(568\) 8.00000i 0.335673i
\(569\) −12.0000 −0.503066 −0.251533 0.967849i \(-0.580935\pi\)
−0.251533 + 0.967849i \(0.580935\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) − 24.0000i − 1.00349i
\(573\) − 8.00000i − 0.334205i
\(574\) −8.00000 −0.333914
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 10.0000i 0.416305i 0.978096 + 0.208153i \(0.0667451\pi\)
−0.978096 + 0.208153i \(0.933255\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 2.00000i 0.0829027i
\(583\) 40.0000i 1.65663i
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) 22.0000i 0.908037i 0.890992 + 0.454019i \(0.150010\pi\)
−0.890992 + 0.454019i \(0.849990\pi\)
\(588\) − 3.00000i − 0.123718i
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) −22.0000 −0.904959
\(592\) 10.0000i 0.410997i
\(593\) 8.00000i 0.328521i 0.986417 + 0.164260i \(0.0525237\pi\)
−0.986417 + 0.164260i \(0.947476\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) 4.00000i 0.163709i
\(598\) − 24.0000i − 0.981433i
\(599\) 28.0000 1.14405 0.572024 0.820237i \(-0.306158\pi\)
0.572024 + 0.820237i \(0.306158\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 24.0000i 0.978167i
\(603\) − 12.0000i − 0.488678i
\(604\) 2.00000 0.0813788
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) − 28.0000i − 1.13648i −0.822861 0.568242i \(-0.807624\pi\)
0.822861 0.568242i \(-0.192376\pi\)
\(608\) 1.00000i 0.0405554i
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 4.00000i 0.161690i
\(613\) 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
\(616\) 8.00000 0.322329
\(617\) − 48.0000i − 1.93241i −0.257780 0.966204i \(-0.582991\pi\)
0.257780 0.966204i \(-0.417009\pi\)
\(618\) 12.0000i 0.482711i
\(619\) −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
\(622\) − 16.0000i − 0.641542i
\(623\) − 16.0000i − 0.641026i
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) − 4.00000i − 0.159745i
\(628\) − 12.0000i − 0.478852i
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) − 10.0000i − 0.397779i
\(633\) − 8.00000i − 0.317971i
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) 18.0000i 0.713186i
\(638\) 24.0000i 0.950169i
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −20.0000 −0.789953 −0.394976 0.918691i \(-0.629247\pi\)
−0.394976 + 0.918691i \(0.629247\pi\)
\(642\) 8.00000i 0.315735i
\(643\) 20.0000i 0.788723i 0.918955 + 0.394362i \(0.129034\pi\)
−0.918955 + 0.394362i \(0.870966\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −40.0000 −1.57014
\(650\) 0 0
\(651\) −12.0000 −0.470317
\(652\) − 16.0000i − 0.626608i
\(653\) 6.00000i 0.234798i 0.993085 + 0.117399i \(0.0374557\pi\)
−0.993085 + 0.117399i \(0.962544\pi\)
\(654\) 16.0000 0.625650
\(655\) 0 0
\(656\) 4.00000 0.156174
\(657\) − 2.00000i − 0.0780274i
\(658\) − 8.00000i − 0.311872i
\(659\) −22.0000 −0.856998 −0.428499 0.903542i \(-0.640958\pi\)
−0.428499 + 0.903542i \(0.640958\pi\)
\(660\) 0 0
\(661\) −8.00000 −0.311164 −0.155582 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) 12.0000i 0.466393i
\(663\) − 24.0000i − 0.932083i
\(664\) 2.00000 0.0776151
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 24.0000i 0.929284i
\(668\) − 24.0000i − 0.928588i
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) − 2.00000i − 0.0771517i
\(673\) − 26.0000i − 1.00223i −0.865382 0.501113i \(-0.832924\pi\)
0.865382 0.501113i \(-0.167076\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) 14.0000i 0.537667i
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 24.0000i 0.919007i
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 26.0000i 0.991962i
\(688\) − 12.0000i − 0.457496i
\(689\) −60.0000 −2.28582
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) − 18.0000i − 0.684257i
\(693\) 8.00000i 0.303895i
\(694\) −2.00000 −0.0759190
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 16.0000i 0.606043i
\(698\) 22.0000i 0.832712i
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 10.0000 0.377695 0.188847 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) − 6.00000i − 0.226455i
\(703\) − 10.0000i − 0.377157i
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 0 0
\(707\) 20.0000i 0.752177i
\(708\) 10.0000i 0.375823i
\(709\) −14.0000 −0.525781 −0.262891 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 8.00000i 0.299813i
\(713\) 24.0000i 0.898807i
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) −2.00000 −0.0747435
\(717\) 0 0
\(718\) 0 0
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) − 1.00000i − 0.0372161i
\(723\) 10.0000i 0.371904i
\(724\) 0 0
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) 26.0000i 0.964287i 0.876092 + 0.482143i \(0.160142\pi\)
−0.876092 + 0.482143i \(0.839858\pi\)
\(728\) 12.0000i 0.444750i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) − 2.00000i − 0.0739221i
\(733\) 8.00000i 0.295487i 0.989026 + 0.147743i \(0.0472010\pi\)
−0.989026 + 0.147743i \(0.952799\pi\)
\(734\) −22.0000 −0.812035
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 48.0000i 1.76810i
\(738\) 4.00000i 0.147242i
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 6.00000 0.220416
\(742\) − 20.0000i − 0.734223i
\(743\) − 48.0000i − 1.76095i −0.474093 0.880475i \(-0.657224\pi\)
0.474093 0.880475i \(-0.342776\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) 2.00000i 0.0731762i
\(748\) − 16.0000i − 0.585018i
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) −38.0000 −1.38664 −0.693320 0.720630i \(-0.743853\pi\)
−0.693320 + 0.720630i \(0.743853\pi\)
\(752\) 4.00000i 0.145865i
\(753\) 0 0
\(754\) −36.0000 −1.31104
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 8.00000i 0.290765i 0.989376 + 0.145382i \(0.0464413\pi\)
−0.989376 + 0.145382i \(0.953559\pi\)
\(758\) 8.00000i 0.290573i
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) −34.0000 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(762\) − 20.0000i − 0.724524i
\(763\) − 32.0000i − 1.15848i
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) − 60.0000i − 2.16647i
\(768\) 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
\(771\) 18.0000 0.648254
\(772\) 14.0000i 0.503871i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 20.0000i 0.717496i
\(778\) − 10.0000i − 0.358517i
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) − 16.0000i − 0.572159i
\(783\) 6.00000i 0.214423i
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) − 22.0000i − 0.783718i
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) 28.0000 0.995565
\(792\) − 4.00000i − 0.142134i
\(793\) 12.0000i 0.426132i
\(794\) −24.0000 −0.851728
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 2.00000i 0.0708436i 0.999372 + 0.0354218i \(0.0112775\pi\)
−0.999372 + 0.0354218i \(0.988723\pi\)
\(798\) 2.00000i 0.0707992i
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) 24.0000i 0.847469i
\(803\) 8.00000i 0.282314i
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) −36.0000 −1.26805
\(807\) − 14.0000i − 0.492823i
\(808\) − 10.0000i − 0.351799i
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) − 12.0000i − 0.421117i
\(813\) 20.0000i 0.701431i
\(814\) 40.0000 1.40200
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) 12.0000i 0.419827i
\(818\) − 6.00000i − 0.209785i
\(819\) −12.0000 −0.419314
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) 26.0000i 0.906303i 0.891434 + 0.453152i \(0.149700\pi\)
−0.891434 + 0.453152i \(0.850300\pi\)
\(824\) −12.0000 −0.418040
\(825\) 0 0
\(826\) 20.0000 0.695889
\(827\) 48.0000i 1.66912i 0.550914 + 0.834562i \(0.314279\pi\)
−0.550914 + 0.834562i \(0.685721\pi\)
\(828\) − 4.00000i − 0.139010i
\(829\) 40.0000 1.38926 0.694629 0.719368i \(-0.255569\pi\)
0.694629 + 0.719368i \(0.255569\pi\)
\(830\) 0 0
\(831\) −28.0000 −0.971309
\(832\) − 6.00000i − 0.208013i
\(833\) 12.0000i 0.415775i
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) 6.00000i 0.207390i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 8.00000i 0.275698i
\(843\) 4.00000i 0.137767i
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) −4.00000 −0.137523
\(847\) − 10.0000i − 0.343604i
\(848\) 10.0000i 0.343401i
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) 40.0000 1.37118
\(852\) − 8.00000i − 0.274075i
\(853\) 36.0000i 1.23262i 0.787505 + 0.616308i \(0.211372\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) 30.0000i 1.02478i 0.858753 + 0.512390i \(0.171240\pi\)
−0.858753 + 0.512390i \(0.828760\pi\)
\(858\) 24.0000i 0.819346i
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) 40.0000i 1.36241i
\(863\) − 32.0000i − 1.08929i −0.838666 0.544646i \(-0.816664\pi\)
0.838666 0.544646i \(-0.183336\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −30.0000 −1.01944
\(867\) 1.00000i 0.0339618i
\(868\) − 12.0000i − 0.407307i
\(869\) −40.0000 −1.35691
\(870\) 0 0
\(871\) −72.0000 −2.43963
\(872\) 16.0000i 0.541828i
\(873\) − 2.00000i − 0.0676897i
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) − 34.0000i − 1.14810i −0.818821 0.574049i \(-0.805372\pi\)
0.818821 0.574049i \(-0.194628\pi\)
\(878\) − 38.0000i − 1.28244i
\(879\) 2.00000 0.0674583
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 3.00000i 0.101015i
\(883\) − 36.0000i − 1.21150i −0.795656 0.605748i \(-0.792874\pi\)
0.795656 0.605748i \(-0.207126\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) −6.00000 −0.201574
\(887\) 48.0000i 1.61168i 0.592132 + 0.805841i \(0.298286\pi\)
−0.592132 + 0.805841i \(0.701714\pi\)
\(888\) − 10.0000i − 0.335578i
\(889\) −40.0000 −1.34156
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 24.0000i 0.803579i
\(893\) − 4.00000i − 0.133855i
\(894\) −14.0000 −0.468230
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 24.0000i 0.801337i
\(898\) 12.0000i 0.400445i
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) −40.0000 −1.33259
\(902\) − 16.0000i − 0.532742i
\(903\) − 24.0000i − 0.798670i
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) −2.00000 −0.0664455
\(907\) − 20.0000i − 0.664089i −0.943264 0.332045i \(-0.892262\pi\)
0.943264 0.332045i \(-0.107738\pi\)
\(908\) − 20.0000i − 0.663723i
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) − 1.00000i − 0.0331133i
\(913\) − 8.00000i − 0.264761i
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) − 24.0000i − 0.792550i
\(918\) − 4.00000i − 0.132020i
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) − 18.0000i − 0.592798i
\(923\) 48.0000i 1.57994i
\(924\) −8.00000 −0.263181
\(925\) 0 0
\(926\) 18.0000 0.591517
\(927\) − 12.0000i − 0.394132i
\(928\) 6.00000i 0.196960i
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) − 12.0000i − 0.393073i
\(933\) 16.0000i 0.523816i
\(934\) −22.0000 −0.719862
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) 26.0000i 0.849383i 0.905338 + 0.424691i \(0.139617\pi\)
−0.905338 + 0.424691i \(0.860383\pi\)
\(938\) − 24.0000i − 0.783628i
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 12.0000i 0.390981i
\(943\) − 16.0000i − 0.521032i
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) − 38.0000i − 1.23483i −0.786636 0.617417i \(-0.788179\pi\)
0.786636 0.617417i \(-0.211821\pi\)
\(948\) 10.0000i 0.324785i
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 8.00000i 0.259281i
\(953\) − 6.00000i − 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) 0 0
\(957\) − 24.0000i − 0.775810i
\(958\) − 24.0000i − 0.775405i
\(959\) 0 0
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 60.0000i 1.93448i
\(963\) − 8.00000i − 0.257796i
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) − 18.0000i − 0.578841i −0.957202 0.289420i \(-0.906537\pi\)
0.957202 0.289420i \(-0.0934626\pi\)
\(968\) 5.00000i 0.160706i
\(969\) 4.00000 0.128499
\(970\) 0 0
\(971\) −42.0000 −1.34784 −0.673922 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 8.00000i − 0.256468i
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) − 42.0000i − 1.34370i −0.740688 0.671850i \(-0.765500\pi\)
0.740688 0.671850i \(-0.234500\pi\)
\(978\) 16.0000i 0.511624i
\(979\) 32.0000 1.02272
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) − 12.0000i − 0.382935i
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) −4.00000 −0.127515
\(985\) 0 0
\(986\) −24.0000 −0.764316
\(987\) 8.00000i 0.254643i
\(988\) 6.00000i 0.190885i
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) −2.00000 −0.0635321 −0.0317660 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 6.00000i 0.190500i
\(993\) − 12.0000i − 0.380808i
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) −2.00000 −0.0633724
\(997\) − 52.0000i − 1.64686i −0.567420 0.823428i \(-0.692059\pi\)
0.567420 0.823428i \(-0.307941\pi\)
\(998\) − 4.00000i − 0.126618i
\(999\) 10.0000 0.316386
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.r.799.1 2
5.2 odd 4 2850.2.a.bb.1.1 1
5.3 odd 4 570.2.a.a.1.1 1
5.4 even 2 inner 2850.2.d.r.799.2 2
15.2 even 4 8550.2.a.n.1.1 1
15.8 even 4 1710.2.a.q.1.1 1
20.3 even 4 4560.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.a.1.1 1 5.3 odd 4
1710.2.a.q.1.1 1 15.8 even 4
2850.2.a.bb.1.1 1 5.2 odd 4
2850.2.d.r.799.1 2 1.1 even 1 trivial
2850.2.d.r.799.2 2 5.4 even 2 inner
4560.2.a.t.1.1 1 20.3 even 4
8550.2.a.n.1.1 1 15.2 even 4