Properties

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

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} +4.00000 q^{21} +4.00000i q^{22} -8.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} -6.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +10.0000i q^{37} -1.00000i q^{38} +2.00000 q^{39} -2.00000 q^{41} -4.00000i q^{42} +12.0000i q^{43} +4.00000 q^{44} -8.00000 q^{46} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} +2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} +4.00000 q^{56} +1.00000i q^{57} +6.00000i q^{58} -10.0000 q^{61} -4.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +4.00000i q^{67} -2.00000i q^{68} +8.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} +10.0000 q^{74} -1.00000 q^{76} +16.0000i q^{77} -2.00000i q^{78} +12.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -8.00000i q^{83} -4.00000 q^{84} +12.0000 q^{86} -6.00000i q^{87} -4.00000i q^{88} -6.00000 q^{89} -8.00000 q^{91} +8.00000i q^{92} +4.00000i q^{93} +1.00000 q^{96} -18.0000i q^{97} +9.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} - 8 q^{14} + 2 q^{16} + 2 q^{19} + 8 q^{21} - 2 q^{24} - 4 q^{26} - 12 q^{29} + 8 q^{31} + 4 q^{34} + 2 q^{36} + 4 q^{39} - 4 q^{41} + 8 q^{44} - 16 q^{46} - 18 q^{49} - 4 q^{51} - 2 q^{54} + 8 q^{56} - 20 q^{61} - 2 q^{64} - 8 q^{66} + 16 q^{69} - 16 q^{71} + 20 q^{74} - 2 q^{76} + 24 q^{79} + 2 q^{81} - 8 q^{84} + 24 q^{86} - 12 q^{89} - 16 q^{91} + 2 q^{96} + 8 q^{99} + O(q^{100}) \)

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
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 4.00000i 0.852803i
\(23\) − 8.00000i − 1.66812i −0.551677 0.834058i \(-0.686012\pi\)
0.551677 0.834058i \(-0.313988\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) − 1.00000i − 0.192450i
\(28\) 4.00000i 0.755929i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 4.00000i − 0.696311i
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) − 4.00000i − 0.617213i
\(43\) 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
\(46\) −8.00000 −1.17954
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 2.00000i 0.277350i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 1.00000i 0.132453i
\(58\) 6.00000i 0.787839i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 4.00000i 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) 8.00000 0.963087
\(70\) 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
\(73\) 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
\(77\) 16.0000i 1.82337i
\(78\) − 2.00000i − 0.226455i
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000i 0.220863i
\(83\) − 8.00000i − 0.878114i −0.898459 0.439057i \(-0.855313\pi\)
0.898459 0.439057i \(-0.144687\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) − 6.00000i − 0.643268i
\(88\) − 4.00000i − 0.426401i
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 8.00000i 0.834058i
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 18.0000i − 1.82762i −0.406138 0.913812i \(-0.633125\pi\)
0.406138 0.913812i \(-0.366875\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 2.00000i 0.198030i
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) − 4.00000i − 0.377964i
\(113\) − 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
\(116\) 6.00000 0.557086
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000i 0.905357i
\(123\) − 2.00000i − 0.180334i
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 4.00000i 0.348155i
\(133\) − 4.00000i − 0.346844i
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) − 14.0000i − 1.19610i −0.801459 0.598050i \(-0.795942\pi\)
0.801459 0.598050i \(-0.204058\pi\)
\(138\) − 8.00000i − 0.681005i
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000i 0.671345i
\(143\) 8.00000i 0.668994i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) − 9.00000i − 0.742307i
\(148\) − 10.0000i − 0.821995i
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) − 2.00000i − 0.161690i
\(154\) 16.0000 1.28932
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 6.00000i 0.478852i 0.970915 + 0.239426i \(0.0769593\pi\)
−0.970915 + 0.239426i \(0.923041\pi\)
\(158\) − 12.0000i − 0.954669i
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −32.0000 −2.52195
\(162\) − 1.00000i − 0.0785674i
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) 16.0000i 1.23812i 0.785345 + 0.619059i \(0.212486\pi\)
−0.785345 + 0.619059i \(0.787514\pi\)
\(168\) 4.00000i 0.308607i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) − 12.0000i − 0.914991i
\(173\) 14.0000i 1.06440i 0.846619 + 0.532200i \(0.178635\pi\)
−0.846619 + 0.532200i \(0.821365\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) 6.00000i 0.449719i
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 8.00000i 0.592999i
\(183\) − 10.0000i − 0.739221i
\(184\) 8.00000 0.589768
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) − 8.00000i − 0.585018i
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 18.0000i 1.29567i 0.761781 + 0.647834i \(0.224325\pi\)
−0.761781 + 0.647834i \(0.775675\pi\)
\(194\) −18.0000 −1.29232
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 10.0000i 0.703598i
\(203\) 24.0000i 1.68447i
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 8.00000i 0.556038i
\(208\) − 2.00000i − 0.138675i
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) − 8.00000i − 0.548151i
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) − 16.0000i − 1.08615i
\(218\) 10.0000i 0.677285i
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 10.0000i 0.671156i
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) − 6.00000i − 0.393919i
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) 12.0000i 0.779484i
\(238\) − 8.00000i − 0.518563i
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 1.00000i 0.0641500i
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) − 2.00000i − 0.127257i
\(248\) 4.00000i 0.254000i
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) − 4.00000i − 0.251976i
\(253\) 32.0000i 2.01182i
\(254\) −8.00000 −0.501965
\(255\) 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
\(259\) 40.0000 2.48548
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 12.0000i 0.741362i
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) − 6.00000i − 0.367194i
\(268\) − 4.00000i − 0.244339i
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 2.00000i 0.121268i
\(273\) − 8.00000i − 0.484182i
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) − 18.0000i − 1.08152i −0.841178 0.540758i \(-0.818138\pi\)
0.841178 0.540758i \(-0.181862\pi\)
\(278\) − 12.0000i − 0.719712i
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 8.00000i 0.472225i
\(288\) 1.00000i 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) − 2.00000i − 0.117041i
\(293\) − 10.0000i − 0.584206i −0.956387 0.292103i \(-0.905645\pi\)
0.956387 0.292103i \(-0.0943550\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) 4.00000i 0.232104i
\(298\) 22.0000i 1.27443i
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) 48.0000 2.76667
\(302\) − 4.00000i − 0.230174i
\(303\) − 10.0000i − 0.574485i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) − 28.0000i − 1.59804i −0.601302 0.799022i \(-0.705351\pi\)
0.601302 0.799022i \(-0.294649\pi\)
\(308\) − 16.0000i − 0.911685i
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 2.00000i 0.113228i
\(313\) − 30.0000i − 1.69570i −0.530236 0.847850i \(-0.677897\pi\)
0.530236 0.847850i \(-0.322103\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) − 30.0000i − 1.68497i −0.538721 0.842484i \(-0.681092\pi\)
0.538721 0.842484i \(-0.318908\pi\)
\(318\) 6.00000i 0.336463i
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 32.0000i 1.78329i
\(323\) 2.00000i 0.111283i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) − 10.0000i − 0.553001i
\(328\) − 2.00000i − 0.110432i
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 8.00000i 0.439057i
\(333\) − 10.0000i − 0.547997i
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) − 34.0000i − 1.85210i −0.377403 0.926049i \(-0.623183\pi\)
0.377403 0.926049i \(-0.376817\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 1.00000i 0.0540738i
\(343\) 8.00000i 0.431959i
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) − 24.0000i − 1.28839i −0.764862 0.644194i \(-0.777193\pi\)
0.764862 0.644194i \(-0.222807\pi\)
\(348\) 6.00000i 0.321634i
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 4.00000i 0.213201i
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 8.00000i 0.423405i
\(358\) 0 0
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 2.00000i − 0.105118i
\(363\) 5.00000i 0.262432i
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 12.0000i 0.626395i 0.949688 + 0.313197i \(0.101400\pi\)
−0.949688 + 0.313197i \(0.898600\pi\)
\(368\) − 8.00000i − 0.417029i
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) − 4.00000i − 0.207390i
\(373\) 30.0000i 1.55334i 0.629907 + 0.776671i \(0.283093\pi\)
−0.629907 + 0.776671i \(0.716907\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 4.00000i 0.205738i
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 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
\(386\) 18.0000 0.916176
\(387\) − 12.0000i − 0.609994i
\(388\) 18.0000i 0.913812i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) − 9.00000i − 0.454569i
\(393\) − 12.0000i − 0.605320i
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) − 2.00000i − 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 16.0000i 0.802008i
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 4.00000i 0.199502i
\(403\) − 8.00000i − 0.398508i
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) − 40.0000i − 1.98273i
\(408\) − 2.00000i − 0.0990148i
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) − 16.0000i − 0.788263i
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 12.0000i 0.587643i
\(418\) 4.00000i 0.195646i
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 40.0000i 1.93574i
\(428\) − 12.0000i − 0.580042i
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 2.00000i 0.0961139i 0.998845 + 0.0480569i \(0.0153029\pi\)
−0.998845 + 0.0480569i \(0.984697\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) − 8.00000i − 0.382692i
\(438\) 2.00000i 0.0955637i
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) − 4.00000i − 0.190261i
\(443\) − 24.0000i − 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) − 22.0000i − 1.04056i
\(448\) 4.00000i 0.188982i
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 6.00000i 0.282216i
\(453\) 4.00000i 0.187936i
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 6.00000i 0.280668i 0.990104 + 0.140334i \(0.0448177\pi\)
−0.990104 + 0.140334i \(0.955182\pi\)
\(458\) 22.0000i 1.02799i
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 16.0000i 0.744387i
\(463\) − 36.0000i − 1.67306i −0.547920 0.836531i \(-0.684580\pi\)
0.547920 0.836531i \(-0.315420\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) − 8.00000i − 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 0 0
\(473\) − 48.0000i − 2.20704i
\(474\) 12.0000 0.551178
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) − 6.00000i − 0.274721i
\(478\) − 8.00000i − 0.365911i
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) − 2.00000i − 0.0910975i
\(483\) − 32.0000i − 1.45605i
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 32.0000i 1.45006i 0.688718 + 0.725029i \(0.258174\pi\)
−0.688718 + 0.725029i \(0.741826\pi\)
\(488\) − 10.0000i − 0.452679i
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) − 12.0000i − 0.540453i
\(494\) −2.00000 −0.0899843
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 32.0000i 1.43540i
\(498\) − 8.00000i − 0.358489i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) 28.0000i 1.24970i
\(503\) 32.0000i 1.42681i 0.700752 + 0.713405i \(0.252848\pi\)
−0.700752 + 0.713405i \(0.747152\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 32.0000 1.42257
\(507\) 9.00000i 0.399704i
\(508\) 8.00000i 0.354943i
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) − 1.00000i − 0.0441942i
\(513\) − 1.00000i − 0.0441511i
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) 0 0
\(518\) − 40.0000i − 1.75750i
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) − 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
\(526\) 24.0000 1.04645
\(527\) 8.00000i 0.348485i
\(528\) − 4.00000i − 0.174078i
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 0 0
\(532\) 4.00000i 0.173422i
\(533\) 4.00000i 0.173259i
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 6.00000i 0.258678i
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 24.0000i 1.03089i
\(543\) 2.00000i 0.0858282i
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) 14.0000i 0.598050i
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 8.00000i 0.340503i
\(553\) − 48.0000i − 2.04117i
\(554\) −18.0000 −0.764747
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) − 30.0000i − 1.27114i −0.772043 0.635570i \(-0.780765\pi\)
0.772043 0.635570i \(-0.219235\pi\)
\(558\) 4.00000i 0.169334i
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) − 30.0000i − 1.26547i
\(563\) − 20.0000i − 0.842900i −0.906852 0.421450i \(-0.861521\pi\)
0.906852 0.421450i \(-0.138479\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) − 4.00000i − 0.167984i
\(568\) − 8.00000i − 0.335673i
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) − 8.00000i − 0.334497i
\(573\) 0 0
\(574\) 8.00000 0.333914
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) −18.0000 −0.748054
\(580\) 0 0
\(581\) −32.0000 −1.32758
\(582\) − 18.0000i − 0.746124i
\(583\) − 24.0000i − 0.993978i
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −10.0000 −0.413096
\(587\) 40.0000i 1.65098i 0.564419 + 0.825488i \(0.309100\pi\)
−0.564419 + 0.825488i \(0.690900\pi\)
\(588\) 9.00000i 0.371154i
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 10.0000i 0.410997i
\(593\) − 10.0000i − 0.410651i −0.978694 0.205325i \(-0.934175\pi\)
0.978694 0.205325i \(-0.0658253\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) − 16.0000i − 0.654836i
\(598\) 16.0000i 0.654289i
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) − 48.0000i − 1.95633i
\(603\) − 4.00000i − 0.162893i
\(604\) −4.00000 −0.162758
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) − 24.0000i − 0.974130i −0.873366 0.487065i \(-0.838067\pi\)
0.873366 0.487065i \(-0.161933\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 0 0
\(612\) 2.00000i 0.0808452i
\(613\) − 6.00000i − 0.242338i −0.992632 0.121169i \(-0.961336\pi\)
0.992632 0.121169i \(-0.0386643\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) −16.0000 −0.644658
\(617\) − 30.0000i − 1.20775i −0.797077 0.603877i \(-0.793622\pi\)
0.797077 0.603877i \(-0.206378\pi\)
\(618\) 16.0000i 0.643614i
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) − 24.0000i − 0.962312i
\(623\) 24.0000i 0.961540i
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −30.0000 −1.19904
\(627\) − 4.00000i − 0.159745i
\(628\) − 6.00000i − 0.239426i
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 12.0000i 0.477334i
\(633\) − 12.0000i − 0.476957i
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 18.0000i 0.713186i
\(638\) − 24.0000i − 0.950169i
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) 12.0000i 0.473602i
\(643\) − 28.0000i − 1.10421i −0.833774 0.552106i \(-0.813824\pi\)
0.833774 0.552106i \(-0.186176\pi\)
\(644\) 32.0000 1.26098
\(645\) 0 0
\(646\) 2.00000 0.0786889
\(647\) 32.0000i 1.25805i 0.777385 + 0.629025i \(0.216546\pi\)
−0.777385 + 0.629025i \(0.783454\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 4.00000i 0.156652i
\(653\) 14.0000i 0.547862i 0.961749 + 0.273931i \(0.0883240\pi\)
−0.961749 + 0.273931i \(0.911676\pi\)
\(654\) −10.0000 −0.391031
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) − 2.00000i − 0.0780274i
\(658\) 0 0
\(659\) 16.0000 0.623272 0.311636 0.950202i \(-0.399123\pi\)
0.311636 + 0.950202i \(0.399123\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 12.0000i 0.466393i
\(663\) 4.00000i 0.155347i
\(664\) 8.00000 0.310460
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 48.0000i 1.85857i
\(668\) − 16.0000i − 0.619059i
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) − 4.00000i − 0.154303i
\(673\) − 14.0000i − 0.539660i −0.962908 0.269830i \(-0.913032\pi\)
0.962908 0.269830i \(-0.0869676\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) − 6.00000i − 0.230429i
\(679\) −72.0000 −2.76311
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 16.0000i 0.612672i
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) − 22.0000i − 0.839352i
\(688\) 12.0000i 0.457496i
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) − 14.0000i − 0.532200i
\(693\) − 16.0000i − 0.607790i
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) − 4.00000i − 0.151511i
\(698\) − 34.0000i − 1.28692i
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 2.00000i 0.0754851i
\(703\) 10.0000i 0.377157i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 40.0000i 1.50435i
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) − 6.00000i − 0.224860i
\(713\) − 32.0000i − 1.19841i
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) 0 0
\(717\) 8.00000i 0.298765i
\(718\) 32.0000i 1.19423i
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 0 0
\(721\) 64.0000 2.38348
\(722\) − 1.00000i − 0.0372161i
\(723\) 2.00000i 0.0743808i
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) − 12.0000i − 0.445055i −0.974926 0.222528i \(-0.928569\pi\)
0.974926 0.222528i \(-0.0714308\pi\)
\(728\) − 8.00000i − 0.296500i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 10.0000i 0.369611i
\(733\) − 14.0000i − 0.517102i −0.965998 0.258551i \(-0.916755\pi\)
0.965998 0.258551i \(-0.0832450\pi\)
\(734\) 12.0000 0.442928
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) − 16.0000i − 0.589368i
\(738\) − 2.00000i − 0.0736210i
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) − 24.0000i − 0.881068i
\(743\) 48.0000i 1.76095i 0.474093 + 0.880475i \(0.342776\pi\)
−0.474093 + 0.880475i \(0.657224\pi\)
\(744\) −4.00000 −0.146647
\(745\) 0 0
\(746\) 30.0000 1.09838
\(747\) 8.00000i 0.292705i
\(748\) 8.00000i 0.292509i
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) 52.0000 1.89751 0.948753 0.316017i \(-0.102346\pi\)
0.948753 + 0.316017i \(0.102346\pi\)
\(752\) 0 0
\(753\) − 28.0000i − 1.02038i
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 38.0000i 1.38113i 0.723269 + 0.690567i \(0.242639\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(758\) − 20.0000i − 0.726433i
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) − 8.00000i − 0.289809i
\(763\) 40.0000i 1.44810i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 1.00000i 0.0360844i
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) − 18.0000i − 0.647834i
\(773\) − 26.0000i − 0.935155i −0.883952 0.467578i \(-0.845127\pi\)
0.883952 0.467578i \(-0.154873\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) 18.0000 0.646162
\(777\) 40.0000i 1.43499i
\(778\) 6.00000i 0.215110i
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) − 16.0000i − 0.572159i
\(783\) 6.00000i 0.214423i
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) − 52.0000i − 1.85360i −0.375555 0.926800i \(-0.622548\pi\)
0.375555 0.926800i \(-0.377452\pi\)
\(788\) − 18.0000i − 0.641223i
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 4.00000i 0.142134i
\(793\) 20.0000i 0.710221i
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) 2.00000i 0.0708436i 0.999372 + 0.0354218i \(0.0112775\pi\)
−0.999372 + 0.0354218i \(0.988723\pi\)
\(798\) − 4.00000i − 0.141598i
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 18.0000i 0.635602i
\(803\) − 8.00000i − 0.282314i
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) − 6.00000i − 0.211210i
\(808\) − 10.0000i − 0.351799i
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) − 24.0000i − 0.842235i
\(813\) − 24.0000i − 0.841717i
\(814\) −40.0000 −1.40200
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 12.0000i 0.419827i
\(818\) − 6.00000i − 0.209785i
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) − 14.0000i − 0.488306i
\(823\) 52.0000i 1.81261i 0.422628 + 0.906303i \(0.361108\pi\)
−0.422628 + 0.906303i \(0.638892\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) − 52.0000i − 1.80822i −0.427303 0.904109i \(-0.640536\pi\)
0.427303 0.904109i \(-0.359464\pi\)
\(828\) − 8.00000i − 0.278019i
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) 2.00000i 0.0693375i
\(833\) − 18.0000i − 0.623663i
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) − 4.00000i − 0.138260i
\(838\) 12.0000i 0.414533i
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 14.0000i 0.482472i
\(843\) 30.0000i 1.03325i
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) − 20.0000i − 0.687208i
\(848\) 6.00000i 0.206041i
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 80.0000 2.74236
\(852\) 8.00000i 0.274075i
\(853\) − 46.0000i − 1.57501i −0.616308 0.787505i \(-0.711372\pi\)
0.616308 0.787505i \(-0.288628\pi\)
\(854\) 40.0000 1.36877
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) − 42.0000i − 1.43469i −0.696717 0.717346i \(-0.745357\pi\)
0.696717 0.717346i \(-0.254643\pi\)
\(858\) 8.00000i 0.273115i
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) 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
\(866\) 2.00000 0.0679628
\(867\) 13.0000i 0.441503i
\(868\) 16.0000i 0.543075i
\(869\) −48.0000 −1.62829
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) − 10.0000i − 0.338643i
\(873\) 18.0000i 0.609208i
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) − 6.00000i − 0.202606i −0.994856 0.101303i \(-0.967699\pi\)
0.994856 0.101303i \(-0.0323011\pi\)
\(878\) − 4.00000i − 0.134993i
\(879\) 10.0000 0.337292
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) − 9.00000i − 0.303046i
\(883\) 4.00000i 0.134611i 0.997732 + 0.0673054i \(0.0214402\pi\)
−0.997732 + 0.0673054i \(0.978560\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) − 48.0000i − 1.61168i −0.592132 0.805841i \(-0.701714\pi\)
0.592132 0.805841i \(-0.298286\pi\)
\(888\) − 10.0000i − 0.335578i
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) − 16.0000i − 0.535720i
\(893\) 0 0
\(894\) −22.0000 −0.735790
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) − 16.0000i − 0.534224i
\(898\) 6.00000i 0.200223i
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) − 8.00000i − 0.266371i
\(903\) 48.0000i 1.59734i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 4.00000 0.132891
\(907\) 20.0000i 0.664089i 0.943264 + 0.332045i \(0.107738\pi\)
−0.943264 + 0.332045i \(0.892262\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) 32.0000i 1.05905i
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 48.0000i 1.58510i
\(918\) − 2.00000i − 0.0660098i
\(919\) 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
\(922\) − 14.0000i − 0.461065i
\(923\) 16.0000i 0.526646i
\(924\) 16.0000 0.526361
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) − 16.0000i − 0.525509i
\(928\) 6.00000i 0.196960i
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −9.00000 −0.294963
\(932\) − 6.00000i − 0.196537i
\(933\) 24.0000i 0.785725i
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 38.0000i 1.24141i 0.784046 + 0.620703i \(0.213153\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(938\) − 16.0000i − 0.522419i
\(939\) 30.0000 0.979013
\(940\) 0 0
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 6.00000i 0.195491i
\(943\) 16.0000i 0.521032i
\(944\) 0 0
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) 40.0000i 1.29983i 0.760009 + 0.649913i \(0.225195\pi\)
−0.760009 + 0.649913i \(0.774805\pi\)
\(948\) − 12.0000i − 0.389742i
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 8.00000i 0.259281i
\(953\) 10.0000i 0.323932i 0.986796 + 0.161966i \(0.0517835\pi\)
−0.986796 + 0.161966i \(0.948217\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 24.0000i 0.775810i
\(958\) − 16.0000i − 0.516937i
\(959\) −56.0000 −1.80833
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) − 20.0000i − 0.644826i
\(963\) − 12.0000i − 0.386695i
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) −32.0000 −1.02958
\(967\) − 52.0000i − 1.67221i −0.548572 0.836104i \(-0.684828\pi\)
0.548572 0.836104i \(-0.315172\pi\)
\(968\) 5.00000i 0.160706i
\(969\) −2.00000 −0.0642493
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 48.0000i − 1.53881i
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) − 50.0000i − 1.59964i −0.600239 0.799821i \(-0.704928\pi\)
0.600239 0.799821i \(-0.295072\pi\)
\(978\) − 4.00000i − 0.127906i
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) − 28.0000i − 0.893516i
\(983\) 48.0000i 1.53096i 0.643458 + 0.765481i \(0.277499\pi\)
−0.643458 + 0.765481i \(0.722501\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 2.00000i 0.0636285i
\(989\) 96.0000 3.05262
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) − 12.0000i − 0.380808i
\(994\) 32.0000 1.01498
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) 14.0000i 0.443384i 0.975117 + 0.221692i \(0.0711580\pi\)
−0.975117 + 0.221692i \(0.928842\pi\)
\(998\) − 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.k.799.1 2
5.2 odd 4 570.2.a.m.1.1 1
5.3 odd 4 2850.2.a.a.1.1 1
5.4 even 2 inner 2850.2.d.k.799.2 2
15.2 even 4 1710.2.a.f.1.1 1
15.8 even 4 8550.2.a.t.1.1 1
20.7 even 4 4560.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.m.1.1 1 5.2 odd 4
1710.2.a.f.1.1 1 15.2 even 4
2850.2.a.a.1.1 1 5.3 odd 4
2850.2.d.k.799.1 2 1.1 even 1 trivial
2850.2.d.k.799.2 2 5.4 even 2 inner
4560.2.a.k.1.1 1 20.7 even 4
8550.2.a.t.1.1 1 15.8 even 4