Properties

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

$q$-expansion

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

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 \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) − 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 \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 4.00000i − 0.970143i −0.874475 0.485071i \(-0.838794\pi\)
0.874475 0.485071i \(-0.161206\pi\)
\(18\) − 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 \(0.136929\pi\)
−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
\(29\) −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 \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) − 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 \(0.367803\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) − 4.00000i − 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) − 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 \(-0.759014\pi\)
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
\(58\) − 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 \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 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 \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 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 \(0.0350096\pi\)
−0.993958 + 0.109764i \(0.964990\pi\)
\(84\) −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 \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 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 \(-0.798655\pi\)
0.806527 0.591198i \(-0.201345\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) − 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 \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) −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 \(0.347461\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) − 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 \(-0.841053\pi\)
0.877896 0.478852i \(-0.158947\pi\)
\(158\) − 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 \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) − 24.0000i − 1.85718i −0.371113 0.928588i \(-0.621024\pi\)
0.371113 0.928588i \(-0.378976\pi\)
\(168\) − 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 \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) −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 \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) − 22.0000i − 1.56744i −0.621117 0.783718i \(-0.713321\pi\)
0.621117 0.783718i \(-0.286679\pi\)
\(198\) − 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 \(0.297074\pi\)
−0.595198 + 0.803579i \(0.702926\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) − 20.0000i − 1.32745i −0.747978 0.663723i \(-0.768975\pi\)
0.747978 0.663723i \(-0.231025\pi\)
\(228\) − 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 \(-0.871412\pi\)
0.919507 0.393073i \(-0.128588\pi\)
\(234\) −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 \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 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 \(0.120634\pi\)
−0.929041 + 0.369976i \(0.879366\pi\)
\(264\) −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 \(-0.681862\pi\)
0.540758 0.841178i \(-0.318138\pi\)
\(278\) 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 \(0.312927\pi\)
−0.554455 + 0.832214i \(0.687073\pi\)
\(284\) −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 \(0.0186065\pi\)
−0.998292 + 0.0584206i \(0.981394\pi\)
\(294\) 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 \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) − 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 \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) 12.0000 0.677199
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) − 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 \(0.376817\pi\)
−0.377403 + 0.926049i \(0.623183\pi\)
\(338\) − 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 \(0.0170960\pi\)
−0.998558 + 0.0536828i \(0.982904\pi\)
\(348\) − 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 \(0.194685\pi\)
−0.818718 + 0.574195i \(0.805315\pi\)
\(368\) 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 \(0.118058\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 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 \(-0.865954\pi\)
0.912633 0.408781i \(-0.134046\pi\)
\(384\) −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 \(0.205734\pi\)
−0.798298 + 0.602263i \(0.794266\pi\)
\(398\) 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 \(0.256250\pi\)
−0.693087 + 0.720854i \(0.743750\pi\)
\(434\) 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 \(0.0455251\pi\)
−0.989790 + 0.142534i \(0.954475\pi\)
\(444\) 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 \(0.172044\pi\)
−0.857455 + 0.514558i \(0.827956\pi\)
\(458\) 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 \(-0.862638\pi\)
0.908325 0.418265i \(-0.137362\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 12.0000 0.555889
\(467\) 22.0000i 1.01804i 0.860755 + 0.509019i \(0.169992\pi\)
−0.860755 + 0.509019i \(0.830008\pi\)
\(468\) − 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 \(-0.850303\pi\)
0.891438 0.453143i \(-0.149697\pi\)
\(488\) − 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 \(-0.942924\pi\)
0.983967 0.178351i \(-0.0570763\pi\)
\(504\) −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 \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) −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 \(0.0272532\pi\)
−0.996337 + 0.0855138i \(0.972747\pi\)
\(548\) 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 \(-0.932049\pi\)
0.977301 0.211857i \(-0.0679510\pi\)
\(558\) 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 \(0.168780\pi\)
−0.862686 + 0.505740i \(0.831220\pi\)
\(564\) 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 \(-0.933255\pi\)
0.978096 0.208153i \(-0.0667451\pi\)
\(578\) 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 \(-0.849990\pi\)
0.890992 0.454019i \(-0.150010\pi\)
\(588\) 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 \(-0.947476\pi\)
0.986417 0.164260i \(-0.0525237\pi\)
\(594\) −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 \(0.192376\pi\)
−0.822861 + 0.568242i \(0.807624\pi\)
\(608\) − 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 \(-0.895269\pi\)
0.946359 0.323117i \(-0.104731\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) 48.0000i 1.93241i 0.257780 + 0.966204i \(0.417009\pi\)
−0.257780 + 0.966204i \(0.582991\pi\)
\(618\) − 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 \(-0.870966\pi\)
0.918955 0.394362i \(-0.129034\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −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 \(-0.962544\pi\)
0.993085 0.117399i \(-0.0374557\pi\)
\(654\) 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 \(0.167076\pi\)
−0.865382 + 0.501113i \(0.832924\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) − 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 \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) −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 \(-0.839858\pi\)
0.876092 0.482143i \(-0.160142\pi\)
\(728\) − 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 \(-0.952799\pi\)
0.989026 0.147743i \(-0.0472010\pi\)
\(734\) −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 \(0.342776\pi\)
−0.474093 + 0.880475i \(0.657224\pi\)
\(744\) 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 \(-0.953559\pi\)
0.989376 0.145382i \(-0.0464413\pi\)
\(758\) − 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 \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) 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 \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) 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 \(-0.988723\pi\)
0.999372 0.0354218i \(-0.0112775\pi\)
\(798\) − 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 \(-0.850300\pi\)
0.891434 0.453152i \(-0.149700\pi\)
\(824\) −12.0000 −0.418040
\(825\) 0 0
\(826\) 20.0000 0.695889
\(827\) − 48.0000i − 1.66912i −0.550914 0.834562i \(-0.685721\pi\)
0.550914 0.834562i \(-0.314279\pi\)
\(828\) 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 \(-0.788628\pi\)
0.787505 0.616308i \(-0.211372\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) − 30.0000i − 1.02478i −0.858753 0.512390i \(-0.828760\pi\)
0.858753 0.512390i \(-0.171240\pi\)
\(858\) − 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 \(0.183336\pi\)
−0.838666 + 0.544646i \(0.816664\pi\)
\(864\) −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 \(0.194628\pi\)
−0.818821 + 0.574049i \(0.805372\pi\)
\(878\) 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 \(0.207126\pi\)
−0.795656 + 0.605748i \(0.792874\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) −6.00000 −0.201574
\(887\) − 48.0000i − 1.61168i −0.592132 0.805841i \(-0.701714\pi\)
0.592132 0.805841i \(-0.298286\pi\)
\(888\) 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 \(0.107738\pi\)
−0.943264 + 0.332045i \(0.892262\pi\)
\(908\) 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 \(-0.860383\pi\)
0.905338 0.424691i \(-0.139617\pi\)
\(938\) 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 \(0.211821\pi\)
−0.786636 + 0.617417i \(0.788179\pi\)
\(948\) − 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 \(0.0309821\pi\)
−0.995267 + 0.0971795i \(0.969018\pi\)
\(954\) −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 \(0.0934626\pi\)
−0.957202 + 0.289420i \(0.906537\pi\)
\(968\) − 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 \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) − 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 \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) −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 \(0.307941\pi\)
−0.567420 + 0.823428i \(0.692059\pi\)
\(998\) 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.799.2 2
5.2 odd 4 570.2.a.a.1.1 1
5.3 odd 4 2850.2.a.bb.1.1 1
5.4 even 2 inner 2850.2.d.r.799.1 2
15.2 even 4 1710.2.a.q.1.1 1
15.8 even 4 8550.2.a.n.1.1 1
20.7 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 5.2 odd 4
1710.2.a.q.1.1 1 15.2 even 4
2850.2.a.bb.1.1 1 5.3 odd 4
2850.2.d.r.799.1 2 5.4 even 2 inner
2850.2.d.r.799.2 2 1.1 even 1 trivial
4560.2.a.t.1.1 1 20.7 even 4
8550.2.a.n.1.1 1 15.8 even 4