Properties

Label 5070.2.b.a.1351.1
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
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 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.a.1351.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000i q^{11} +1.00000 q^{12} -3.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -1.00000i q^{18} +5.00000i q^{19} +1.00000i q^{20} +3.00000i q^{21} +1.00000 q^{22} +4.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +3.00000i q^{28} +1.00000 q^{30} -10.0000i q^{31} -1.00000i q^{32} -1.00000i q^{33} -3.00000 q^{35} -1.00000 q^{36} -1.00000i q^{37} +5.00000 q^{38} +1.00000 q^{40} -6.00000i q^{41} +3.00000 q^{42} +2.00000 q^{43} -1.00000i q^{44} -1.00000i q^{45} -4.00000i q^{46} -9.00000i q^{47} -1.00000 q^{48} -2.00000 q^{49} +1.00000i q^{50} -13.0000 q^{53} +1.00000i q^{54} +1.00000 q^{55} +3.00000 q^{56} -5.00000i q^{57} +4.00000i q^{59} -1.00000i q^{60} -2.00000 q^{61} -10.0000 q^{62} -3.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} +12.0000i q^{67} -4.00000 q^{69} +3.00000i q^{70} +2.00000i q^{71} +1.00000i q^{72} -16.0000i q^{73} -1.00000 q^{74} +1.00000 q^{75} -5.00000i q^{76} +3.00000 q^{77} -10.0000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000i q^{83} -3.00000i q^{84} -2.00000i q^{86} -1.00000 q^{88} +1.00000i q^{89} -1.00000 q^{90} -4.00000 q^{92} +10.0000i q^{93} -9.00000 q^{94} +5.00000 q^{95} +1.00000i q^{96} -12.0000i q^{97} +2.00000i q^{98} +1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} - 6 q^{14} + 2 q^{16} + 2 q^{22} + 8 q^{23} - 2 q^{25} - 2 q^{27} + 2 q^{30} - 6 q^{35} - 2 q^{36} + 10 q^{38} + 2 q^{40} + 6 q^{42} + 4 q^{43} - 2 q^{48} - 4 q^{49} - 26 q^{53} + 2 q^{55} + 6 q^{56} - 4 q^{61} - 20 q^{62} - 2 q^{64} - 2 q^{66} - 8 q^{69} - 2 q^{74} + 2 q^{75} + 6 q^{77} - 20 q^{79} + 2 q^{81} - 12 q^{82} - 2 q^{88} - 2 q^{90} - 8 q^{92} - 18 q^{94} + 10 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) 1.00000i 0.408248i
\(7\) − 3.00000i − 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 1.00000i 0.301511i 0.988571 + 0.150756i \(0.0481707\pi\)
−0.988571 + 0.150756i \(0.951829\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −3.00000 −0.801784
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 5.00000i 1.14708i 0.819178 + 0.573539i \(0.194430\pi\)
−0.819178 + 0.573539i \(0.805570\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 3.00000i 0.654654i
\(22\) 1.00000 0.213201
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 3.00000i 0.566947i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 1.00000 0.182574
\(31\) − 10.0000i − 1.79605i −0.439941 0.898027i \(-0.645001\pi\)
0.439941 0.898027i \(-0.354999\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 1.00000i − 0.174078i
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) −1.00000 −0.166667
\(37\) − 1.00000i − 0.164399i −0.996616 0.0821995i \(-0.973806\pi\)
0.996616 0.0821995i \(-0.0261945\pi\)
\(38\) 5.00000 0.811107
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) − 6.00000i − 0.937043i −0.883452 0.468521i \(-0.844787\pi\)
0.883452 0.468521i \(-0.155213\pi\)
\(42\) 3.00000 0.462910
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) − 1.00000i − 0.150756i
\(45\) − 1.00000i − 0.149071i
\(46\) − 4.00000i − 0.589768i
\(47\) − 9.00000i − 1.31278i −0.754420 0.656392i \(-0.772082\pi\)
0.754420 0.656392i \(-0.227918\pi\)
\(48\) −1.00000 −0.144338
\(49\) −2.00000 −0.285714
\(50\) 1.00000i 0.141421i
\(51\) 0 0
\(52\) 0 0
\(53\) −13.0000 −1.78569 −0.892844 0.450367i \(-0.851293\pi\)
−0.892844 + 0.450367i \(0.851293\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 1.00000 0.134840
\(56\) 3.00000 0.400892
\(57\) − 5.00000i − 0.662266i
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −10.0000 −1.27000
\(63\) − 3.00000i − 0.377964i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 3.00000i 0.358569i
\(71\) 2.00000i 0.237356i 0.992933 + 0.118678i \(0.0378657\pi\)
−0.992933 + 0.118678i \(0.962134\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 16.0000i − 1.87266i −0.351123 0.936329i \(-0.614200\pi\)
0.351123 0.936329i \(-0.385800\pi\)
\(74\) −1.00000 −0.116248
\(75\) 1.00000 0.115470
\(76\) − 5.00000i − 0.573539i
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) − 3.00000i − 0.327327i
\(85\) 0 0
\(86\) − 2.00000i − 0.215666i
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 1.00000i 0.106000i 0.998595 + 0.0529999i \(0.0168783\pi\)
−0.998595 + 0.0529999i \(0.983122\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 10.0000i 1.03695i
\(94\) −9.00000 −0.928279
\(95\) 5.00000 0.512989
\(96\) 1.00000i 0.102062i
\(97\) − 12.0000i − 1.21842i −0.793011 0.609208i \(-0.791488\pi\)
0.793011 0.609208i \(-0.208512\pi\)
\(98\) 2.00000i 0.202031i
\(99\) 1.00000i 0.100504i
\(100\) 1.00000 0.100000
\(101\) −4.00000 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 0 0
\(103\) 9.00000 0.886796 0.443398 0.896325i \(-0.353773\pi\)
0.443398 + 0.896325i \(0.353773\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 13.0000i 1.26267i
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.0000i 0.957826i 0.877862 + 0.478913i \(0.158969\pi\)
−0.877862 + 0.478913i \(0.841031\pi\)
\(110\) − 1.00000i − 0.0953463i
\(111\) 1.00000i 0.0949158i
\(112\) − 3.00000i − 0.283473i
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) −5.00000 −0.468293
\(115\) − 4.00000i − 0.373002i
\(116\) 0 0
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 10.0000 0.909091
\(122\) 2.00000i 0.181071i
\(123\) 6.00000i 0.541002i
\(124\) 10.0000i 0.898027i
\(125\) 1.00000i 0.0894427i
\(126\) −3.00000 −0.267261
\(127\) −5.00000 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 1.00000i 0.0870388i
\(133\) 15.0000 1.30066
\(134\) 12.0000 1.03664
\(135\) 1.00000i 0.0860663i
\(136\) 0 0
\(137\) 16.0000i 1.36697i 0.729964 + 0.683486i \(0.239537\pi\)
−0.729964 + 0.683486i \(0.760463\pi\)
\(138\) 4.00000i 0.340503i
\(139\) −9.00000 −0.763370 −0.381685 0.924292i \(-0.624656\pi\)
−0.381685 + 0.924292i \(0.624656\pi\)
\(140\) 3.00000 0.253546
\(141\) 9.00000i 0.757937i
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −16.0000 −1.32417
\(147\) 2.00000 0.164957
\(148\) 1.00000i 0.0821995i
\(149\) 6.00000i 0.491539i 0.969328 + 0.245770i \(0.0790407\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) − 6.00000i − 0.488273i −0.969741 0.244137i \(-0.921495\pi\)
0.969741 0.244137i \(-0.0785045\pi\)
\(152\) −5.00000 −0.405554
\(153\) 0 0
\(154\) − 3.00000i − 0.241747i
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) 1.00000 0.0798087 0.0399043 0.999204i \(-0.487295\pi\)
0.0399043 + 0.999204i \(0.487295\pi\)
\(158\) 10.0000i 0.795557i
\(159\) 13.0000 1.03097
\(160\) −1.00000 −0.0790569
\(161\) − 12.0000i − 0.945732i
\(162\) − 1.00000i − 0.0785674i
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) 6.00000i 0.468521i
\(165\) −1.00000 −0.0778499
\(166\) −12.0000 −0.931381
\(167\) − 13.0000i − 1.00597i −0.864295 0.502985i \(-0.832235\pi\)
0.864295 0.502985i \(-0.167765\pi\)
\(168\) −3.00000 −0.231455
\(169\) 0 0
\(170\) 0 0
\(171\) 5.00000i 0.382360i
\(172\) −2.00000 −0.152499
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 0 0
\(175\) 3.00000i 0.226779i
\(176\) 1.00000i 0.0753778i
\(177\) − 4.00000i − 0.300658i
\(178\) 1.00000 0.0749532
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 4.00000i 0.294884i
\(185\) −1.00000 −0.0735215
\(186\) 10.0000 0.733236
\(187\) 0 0
\(188\) 9.00000i 0.656392i
\(189\) 3.00000i 0.218218i
\(190\) − 5.00000i − 0.362738i
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 1.00000 0.0721688
\(193\) 16.0000i 1.15171i 0.817554 + 0.575853i \(0.195330\pi\)
−0.817554 + 0.575853i \(0.804670\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 15.0000i 1.06871i 0.845262 + 0.534353i \(0.179445\pi\)
−0.845262 + 0.534353i \(0.820555\pi\)
\(198\) 1.00000 0.0710669
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) − 12.0000i − 0.846415i
\(202\) 4.00000i 0.281439i
\(203\) 0 0
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) − 9.00000i − 0.627060i
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) −5.00000 −0.345857
\(210\) − 3.00000i − 0.207020i
\(211\) −15.0000 −1.03264 −0.516321 0.856395i \(-0.672699\pi\)
−0.516321 + 0.856395i \(0.672699\pi\)
\(212\) 13.0000 0.892844
\(213\) − 2.00000i − 0.137038i
\(214\) 6.00000i 0.410152i
\(215\) − 2.00000i − 0.136399i
\(216\) − 1.00000i − 0.0680414i
\(217\) −30.0000 −2.03653
\(218\) 10.0000 0.677285
\(219\) 16.0000i 1.08118i
\(220\) −1.00000 −0.0674200
\(221\) 0 0
\(222\) 1.00000 0.0671156
\(223\) − 11.0000i − 0.736614i −0.929704 0.368307i \(-0.879937\pi\)
0.929704 0.368307i \(-0.120063\pi\)
\(224\) −3.00000 −0.200446
\(225\) −1.00000 −0.0666667
\(226\) − 16.0000i − 1.06430i
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 5.00000i 0.331133i
\(229\) − 10.0000i − 0.660819i −0.943838 0.330409i \(-0.892813\pi\)
0.943838 0.330409i \(-0.107187\pi\)
\(230\) −4.00000 −0.263752
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) − 4.00000i − 0.260378i
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) 2.00000i 0.129369i 0.997906 + 0.0646846i \(0.0206041\pi\)
−0.997906 + 0.0646846i \(0.979396\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 15.0000i 0.966235i 0.875556 + 0.483117i \(0.160496\pi\)
−0.875556 + 0.483117i \(0.839504\pi\)
\(242\) − 10.0000i − 0.642824i
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 2.00000i 0.127775i
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 10.0000 0.635001
\(249\) 12.0000i 0.760469i
\(250\) 1.00000 0.0632456
\(251\) −11.0000 −0.694314 −0.347157 0.937807i \(-0.612853\pi\)
−0.347157 + 0.937807i \(0.612853\pi\)
\(252\) 3.00000i 0.188982i
\(253\) 4.00000i 0.251478i
\(254\) 5.00000i 0.313728i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 24.0000 1.49708 0.748539 0.663090i \(-0.230755\pi\)
0.748539 + 0.663090i \(0.230755\pi\)
\(258\) 2.00000i 0.124515i
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) 0 0
\(262\) 15.0000i 0.926703i
\(263\) 3.00000 0.184988 0.0924940 0.995713i \(-0.470516\pi\)
0.0924940 + 0.995713i \(0.470516\pi\)
\(264\) 1.00000 0.0615457
\(265\) 13.0000i 0.798584i
\(266\) − 15.0000i − 0.919709i
\(267\) − 1.00000i − 0.0611990i
\(268\) − 12.0000i − 0.733017i
\(269\) −20.0000 −1.21942 −0.609711 0.792624i \(-0.708714\pi\)
−0.609711 + 0.792624i \(0.708714\pi\)
\(270\) 1.00000 0.0608581
\(271\) − 24.0000i − 1.45790i −0.684569 0.728948i \(-0.740010\pi\)
0.684569 0.728948i \(-0.259990\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 16.0000 0.966595
\(275\) − 1.00000i − 0.0603023i
\(276\) 4.00000 0.240772
\(277\) −23.0000 −1.38194 −0.690968 0.722885i \(-0.742815\pi\)
−0.690968 + 0.722885i \(0.742815\pi\)
\(278\) 9.00000i 0.539784i
\(279\) − 10.0000i − 0.598684i
\(280\) − 3.00000i − 0.179284i
\(281\) 10.0000i 0.596550i 0.954480 + 0.298275i \(0.0964112\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) 9.00000 0.535942
\(283\) −2.00000 −0.118888 −0.0594438 0.998232i \(-0.518933\pi\)
−0.0594438 + 0.998232i \(0.518933\pi\)
\(284\) − 2.00000i − 0.118678i
\(285\) −5.00000 −0.296174
\(286\) 0 0
\(287\) −18.0000 −1.06251
\(288\) − 1.00000i − 0.0589256i
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 12.0000i 0.703452i
\(292\) 16.0000i 0.936329i
\(293\) 13.0000i 0.759468i 0.925096 + 0.379734i \(0.123985\pi\)
−0.925096 + 0.379734i \(0.876015\pi\)
\(294\) − 2.00000i − 0.116642i
\(295\) 4.00000 0.232889
\(296\) 1.00000 0.0581238
\(297\) − 1.00000i − 0.0580259i
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) − 6.00000i − 0.345834i
\(302\) −6.00000 −0.345261
\(303\) 4.00000 0.229794
\(304\) 5.00000i 0.286770i
\(305\) 2.00000i 0.114520i
\(306\) 0 0
\(307\) 18.0000i 1.02731i 0.857996 + 0.513657i \(0.171710\pi\)
−0.857996 + 0.513657i \(0.828290\pi\)
\(308\) −3.00000 −0.170941
\(309\) −9.00000 −0.511992
\(310\) 10.0000i 0.567962i
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −34.0000 −1.92179 −0.960897 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) − 1.00000i − 0.0564333i
\(315\) −3.00000 −0.169031
\(316\) 10.0000 0.562544
\(317\) − 19.0000i − 1.06715i −0.845754 0.533573i \(-0.820849\pi\)
0.845754 0.533573i \(-0.179151\pi\)
\(318\) − 13.0000i − 0.729004i
\(319\) 0 0
\(320\) 1.00000i 0.0559017i
\(321\) 6.00000 0.334887
\(322\) −12.0000 −0.668734
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) − 10.0000i − 0.553001i
\(328\) 6.00000 0.331295
\(329\) −27.0000 −1.48856
\(330\) 1.00000i 0.0550482i
\(331\) − 28.0000i − 1.53902i −0.638635 0.769510i \(-0.720501\pi\)
0.638635 0.769510i \(-0.279499\pi\)
\(332\) 12.0000i 0.658586i
\(333\) − 1.00000i − 0.0547997i
\(334\) −13.0000 −0.711328
\(335\) 12.0000 0.655630
\(336\) 3.00000i 0.163663i
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) −16.0000 −0.869001
\(340\) 0 0
\(341\) 10.0000 0.541530
\(342\) 5.00000 0.270369
\(343\) − 15.0000i − 0.809924i
\(344\) 2.00000i 0.107833i
\(345\) 4.00000i 0.215353i
\(346\) − 9.00000i − 0.483843i
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 0 0
\(349\) 36.0000i 1.92704i 0.267644 + 0.963518i \(0.413755\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 3.00000 0.160357
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 36.0000i 1.91609i 0.286623 + 0.958043i \(0.407467\pi\)
−0.286623 + 0.958043i \(0.592533\pi\)
\(354\) −4.00000 −0.212598
\(355\) 2.00000 0.106149
\(356\) − 1.00000i − 0.0529999i
\(357\) 0 0
\(358\) 12.0000i 0.634220i
\(359\) 34.0000i 1.79445i 0.441572 + 0.897226i \(0.354421\pi\)
−0.441572 + 0.897226i \(0.645579\pi\)
\(360\) 1.00000 0.0527046
\(361\) −6.00000 −0.315789
\(362\) 6.00000i 0.315353i
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) −16.0000 −0.837478
\(366\) − 2.00000i − 0.104542i
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 4.00000 0.208514
\(369\) − 6.00000i − 0.312348i
\(370\) 1.00000i 0.0519875i
\(371\) 39.0000i 2.02478i
\(372\) − 10.0000i − 0.518476i
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) − 1.00000i − 0.0516398i
\(376\) 9.00000 0.464140
\(377\) 0 0
\(378\) 3.00000 0.154303
\(379\) 1.00000i 0.0513665i 0.999670 + 0.0256833i \(0.00817614\pi\)
−0.999670 + 0.0256833i \(0.991824\pi\)
\(380\) −5.00000 −0.256495
\(381\) 5.00000 0.256158
\(382\) 18.0000i 0.920960i
\(383\) 28.0000i 1.43073i 0.698749 + 0.715367i \(0.253740\pi\)
−0.698749 + 0.715367i \(0.746260\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) − 3.00000i − 0.152894i
\(386\) 16.0000 0.814379
\(387\) 2.00000 0.101666
\(388\) 12.0000i 0.609208i
\(389\) 16.0000 0.811232 0.405616 0.914044i \(-0.367057\pi\)
0.405616 + 0.914044i \(0.367057\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 2.00000i − 0.101015i
\(393\) 15.0000 0.756650
\(394\) 15.0000 0.755689
\(395\) 10.0000i 0.503155i
\(396\) − 1.00000i − 0.0502519i
\(397\) − 33.0000i − 1.65622i −0.560564 0.828111i \(-0.689416\pi\)
0.560564 0.828111i \(-0.310584\pi\)
\(398\) − 2.00000i − 0.100251i
\(399\) −15.0000 −0.750939
\(400\) −1.00000 −0.0500000
\(401\) 25.0000i 1.24844i 0.781248 + 0.624220i \(0.214583\pi\)
−0.781248 + 0.624220i \(0.785417\pi\)
\(402\) −12.0000 −0.598506
\(403\) 0 0
\(404\) 4.00000 0.199007
\(405\) − 1.00000i − 0.0496904i
\(406\) 0 0
\(407\) 1.00000 0.0495682
\(408\) 0 0
\(409\) − 17.0000i − 0.840596i −0.907386 0.420298i \(-0.861926\pi\)
0.907386 0.420298i \(-0.138074\pi\)
\(410\) 6.00000i 0.296319i
\(411\) − 16.0000i − 0.789222i
\(412\) −9.00000 −0.443398
\(413\) 12.0000 0.590481
\(414\) − 4.00000i − 0.196589i
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 9.00000 0.440732
\(418\) 5.00000i 0.244558i
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) −3.00000 −0.146385
\(421\) − 20.0000i − 0.974740i −0.873195 0.487370i \(-0.837956\pi\)
0.873195 0.487370i \(-0.162044\pi\)
\(422\) 15.0000i 0.730189i
\(423\) − 9.00000i − 0.437595i
\(424\) − 13.0000i − 0.631336i
\(425\) 0 0
\(426\) −2.00000 −0.0969003
\(427\) 6.00000i 0.290360i
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) −2.00000 −0.0964486
\(431\) − 36.0000i − 1.73406i −0.498257 0.867029i \(-0.666026\pi\)
0.498257 0.867029i \(-0.333974\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 30.0000i 1.44005i
\(435\) 0 0
\(436\) − 10.0000i − 0.478913i
\(437\) 20.0000i 0.956730i
\(438\) 16.0000 0.764510
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 1.00000i 0.0476731i
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) −18.0000 −0.855206 −0.427603 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) − 1.00000i − 0.0474579i
\(445\) 1.00000 0.0474045
\(446\) −11.0000 −0.520865
\(447\) − 6.00000i − 0.283790i
\(448\) 3.00000i 0.141737i
\(449\) − 15.0000i − 0.707894i −0.935266 0.353947i \(-0.884839\pi\)
0.935266 0.353947i \(-0.115161\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 6.00000 0.282529
\(452\) −16.0000 −0.752577
\(453\) 6.00000i 0.281905i
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 5.00000 0.234146
\(457\) − 22.0000i − 1.02912i −0.857455 0.514558i \(-0.827956\pi\)
0.857455 0.514558i \(-0.172044\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 4.00000i 0.186501i
\(461\) 12.0000i 0.558896i 0.960161 + 0.279448i \(0.0901514\pi\)
−0.960161 + 0.279448i \(0.909849\pi\)
\(462\) 3.00000i 0.139573i
\(463\) − 16.0000i − 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) 0 0
\(465\) 10.0000 0.463739
\(466\) 10.0000i 0.463241i
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) 36.0000 1.66233
\(470\) 9.00000i 0.415139i
\(471\) −1.00000 −0.0460776
\(472\) −4.00000 −0.184115
\(473\) 2.00000i 0.0919601i
\(474\) − 10.0000i − 0.459315i
\(475\) − 5.00000i − 0.229416i
\(476\) 0 0
\(477\) −13.0000 −0.595229
\(478\) 2.00000 0.0914779
\(479\) − 8.00000i − 0.365529i −0.983157 0.182765i \(-0.941495\pi\)
0.983157 0.182765i \(-0.0585046\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 15.0000 0.683231
\(483\) 12.0000i 0.546019i
\(484\) −10.0000 −0.454545
\(485\) −12.0000 −0.544892
\(486\) 1.00000i 0.0453609i
\(487\) − 35.0000i − 1.58600i −0.609221 0.793001i \(-0.708518\pi\)
0.609221 0.793001i \(-0.291482\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) 20.0000i 0.904431i
\(490\) 2.00000 0.0903508
\(491\) −25.0000 −1.12823 −0.564117 0.825695i \(-0.690783\pi\)
−0.564117 + 0.825695i \(0.690783\pi\)
\(492\) − 6.00000i − 0.270501i
\(493\) 0 0
\(494\) 0 0
\(495\) 1.00000 0.0449467
\(496\) − 10.0000i − 0.449013i
\(497\) 6.00000 0.269137
\(498\) 12.0000 0.537733
\(499\) − 20.0000i − 0.895323i −0.894203 0.447661i \(-0.852257\pi\)
0.894203 0.447661i \(-0.147743\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) 13.0000i 0.580797i
\(502\) 11.0000i 0.490954i
\(503\) −1.00000 −0.0445878 −0.0222939 0.999751i \(-0.507097\pi\)
−0.0222939 + 0.999751i \(0.507097\pi\)
\(504\) 3.00000 0.133631
\(505\) 4.00000i 0.177998i
\(506\) 4.00000 0.177822
\(507\) 0 0
\(508\) 5.00000 0.221839
\(509\) − 18.0000i − 0.797836i −0.916987 0.398918i \(-0.869386\pi\)
0.916987 0.398918i \(-0.130614\pi\)
\(510\) 0 0
\(511\) −48.0000 −2.12339
\(512\) − 1.00000i − 0.0441942i
\(513\) − 5.00000i − 0.220755i
\(514\) − 24.0000i − 1.05859i
\(515\) − 9.00000i − 0.396587i
\(516\) 2.00000 0.0880451
\(517\) 9.00000 0.395820
\(518\) 3.00000i 0.131812i
\(519\) −9.00000 −0.395056
\(520\) 0 0
\(521\) 33.0000 1.44576 0.722878 0.690976i \(-0.242819\pi\)
0.722878 + 0.690976i \(0.242819\pi\)
\(522\) 0 0
\(523\) 6.00000 0.262362 0.131181 0.991358i \(-0.458123\pi\)
0.131181 + 0.991358i \(0.458123\pi\)
\(524\) 15.0000 0.655278
\(525\) − 3.00000i − 0.130931i
\(526\) − 3.00000i − 0.130806i
\(527\) 0 0
\(528\) − 1.00000i − 0.0435194i
\(529\) −7.00000 −0.304348
\(530\) 13.0000 0.564684
\(531\) 4.00000i 0.173585i
\(532\) −15.0000 −0.650332
\(533\) 0 0
\(534\) −1.00000 −0.0432742
\(535\) 6.00000i 0.259403i
\(536\) −12.0000 −0.518321
\(537\) 12.0000 0.517838
\(538\) 20.0000i 0.862261i
\(539\) − 2.00000i − 0.0861461i
\(540\) − 1.00000i − 0.0430331i
\(541\) − 2.00000i − 0.0859867i −0.999075 0.0429934i \(-0.986311\pi\)
0.999075 0.0429934i \(-0.0136894\pi\)
\(542\) −24.0000 −1.03089
\(543\) 6.00000 0.257485
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) 34.0000 1.45374 0.726868 0.686778i \(-0.240975\pi\)
0.726868 + 0.686778i \(0.240975\pi\)
\(548\) − 16.0000i − 0.683486i
\(549\) −2.00000 −0.0853579
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) − 4.00000i − 0.170251i
\(553\) 30.0000i 1.27573i
\(554\) 23.0000i 0.977176i
\(555\) 1.00000 0.0424476
\(556\) 9.00000 0.381685
\(557\) − 3.00000i − 0.127114i −0.997978 0.0635570i \(-0.979756\pi\)
0.997978 0.0635570i \(-0.0202445\pi\)
\(558\) −10.0000 −0.423334
\(559\) 0 0
\(560\) −3.00000 −0.126773
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) − 9.00000i − 0.378968i
\(565\) − 16.0000i − 0.673125i
\(566\) 2.00000i 0.0840663i
\(567\) − 3.00000i − 0.125988i
\(568\) −2.00000 −0.0839181
\(569\) 11.0000 0.461144 0.230572 0.973055i \(-0.425940\pi\)
0.230572 + 0.973055i \(0.425940\pi\)
\(570\) 5.00000i 0.209427i
\(571\) −7.00000 −0.292941 −0.146470 0.989215i \(-0.546791\pi\)
−0.146470 + 0.989215i \(0.546791\pi\)
\(572\) 0 0
\(573\) 18.0000 0.751961
\(574\) 18.0000i 0.751305i
\(575\) −4.00000 −0.166812
\(576\) −1.00000 −0.0416667
\(577\) − 18.0000i − 0.749350i −0.927156 0.374675i \(-0.877754\pi\)
0.927156 0.374675i \(-0.122246\pi\)
\(578\) 17.0000i 0.707107i
\(579\) − 16.0000i − 0.664937i
\(580\) 0 0
\(581\) −36.0000 −1.49353
\(582\) 12.0000 0.497416
\(583\) − 13.0000i − 0.538405i
\(584\) 16.0000 0.662085
\(585\) 0 0
\(586\) 13.0000 0.537025
\(587\) 18.0000i 0.742940i 0.928445 + 0.371470i \(0.121146\pi\)
−0.928445 + 0.371470i \(0.878854\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 50.0000 2.06021
\(590\) − 4.00000i − 0.164677i
\(591\) − 15.0000i − 0.617018i
\(592\) − 1.00000i − 0.0410997i
\(593\) 28.0000i 1.14982i 0.818216 + 0.574911i \(0.194963\pi\)
−0.818216 + 0.574911i \(0.805037\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) − 6.00000i − 0.245770i
\(597\) −2.00000 −0.0818546
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) −27.0000 −1.10135 −0.550676 0.834719i \(-0.685630\pi\)
−0.550676 + 0.834719i \(0.685630\pi\)
\(602\) −6.00000 −0.244542
\(603\) 12.0000i 0.488678i
\(604\) 6.00000i 0.244137i
\(605\) − 10.0000i − 0.406558i
\(606\) − 4.00000i − 0.162489i
\(607\) 37.0000 1.50178 0.750892 0.660425i \(-0.229624\pi\)
0.750892 + 0.660425i \(0.229624\pi\)
\(608\) 5.00000 0.202777
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) 0 0
\(612\) 0 0
\(613\) − 23.0000i − 0.928961i −0.885583 0.464481i \(-0.846241\pi\)
0.885583 0.464481i \(-0.153759\pi\)
\(614\) 18.0000 0.726421
\(615\) 6.00000 0.241943
\(616\) 3.00000i 0.120873i
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 9.00000i 0.362033i
\(619\) − 31.0000i − 1.24600i −0.782224 0.622998i \(-0.785915\pi\)
0.782224 0.622998i \(-0.214085\pi\)
\(620\) 10.0000 0.401610
\(621\) −4.00000 −0.160514
\(622\) − 12.0000i − 0.481156i
\(623\) 3.00000 0.120192
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 34.0000i 1.35891i
\(627\) 5.00000 0.199681
\(628\) −1.00000 −0.0399043
\(629\) 0 0
\(630\) 3.00000i 0.119523i
\(631\) − 4.00000i − 0.159237i −0.996825 0.0796187i \(-0.974630\pi\)
0.996825 0.0796187i \(-0.0253703\pi\)
\(632\) − 10.0000i − 0.397779i
\(633\) 15.0000 0.596196
\(634\) −19.0000 −0.754586
\(635\) 5.00000i 0.198419i
\(636\) −13.0000 −0.515484
\(637\) 0 0
\(638\) 0 0
\(639\) 2.00000i 0.0791188i
\(640\) 1.00000 0.0395285
\(641\) 9.00000 0.355479 0.177739 0.984078i \(-0.443122\pi\)
0.177739 + 0.984078i \(0.443122\pi\)
\(642\) − 6.00000i − 0.236801i
\(643\) 44.0000i 1.73519i 0.497271 + 0.867595i \(0.334335\pi\)
−0.497271 + 0.867595i \(0.665665\pi\)
\(644\) 12.0000i 0.472866i
\(645\) 2.00000i 0.0787499i
\(646\) 0 0
\(647\) −9.00000 −0.353827 −0.176913 0.984226i \(-0.556611\pi\)
−0.176913 + 0.984226i \(0.556611\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) 30.0000 1.17579
\(652\) 20.0000i 0.783260i
\(653\) 41.0000 1.60445 0.802227 0.597019i \(-0.203648\pi\)
0.802227 + 0.597019i \(0.203648\pi\)
\(654\) −10.0000 −0.391031
\(655\) 15.0000i 0.586098i
\(656\) − 6.00000i − 0.234261i
\(657\) − 16.0000i − 0.624219i
\(658\) 27.0000i 1.05257i
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 1.00000 0.0389249
\(661\) 14.0000i 0.544537i 0.962221 + 0.272268i \(0.0877739\pi\)
−0.962221 + 0.272268i \(0.912226\pi\)
\(662\) −28.0000 −1.08825
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) − 15.0000i − 0.581675i
\(666\) −1.00000 −0.0387492
\(667\) 0 0
\(668\) 13.0000i 0.502985i
\(669\) 11.0000i 0.425285i
\(670\) − 12.0000i − 0.463600i
\(671\) − 2.00000i − 0.0772091i
\(672\) 3.00000 0.115728
\(673\) 32.0000 1.23351 0.616755 0.787155i \(-0.288447\pi\)
0.616755 + 0.787155i \(0.288447\pi\)
\(674\) 2.00000i 0.0770371i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −26.0000 −0.999261 −0.499631 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) 16.0000i 0.614476i
\(679\) −36.0000 −1.38155
\(680\) 0 0
\(681\) − 20.0000i − 0.766402i
\(682\) − 10.0000i − 0.382920i
\(683\) − 26.0000i − 0.994862i −0.867503 0.497431i \(-0.834277\pi\)
0.867503 0.497431i \(-0.165723\pi\)
\(684\) − 5.00000i − 0.191180i
\(685\) 16.0000 0.611329
\(686\) −15.0000 −0.572703
\(687\) 10.0000i 0.381524i
\(688\) 2.00000 0.0762493
\(689\) 0 0
\(690\) 4.00000 0.152277
\(691\) − 33.0000i − 1.25538i −0.778464 0.627690i \(-0.784001\pi\)
0.778464 0.627690i \(-0.215999\pi\)
\(692\) −9.00000 −0.342129
\(693\) 3.00000 0.113961
\(694\) 28.0000i 1.06287i
\(695\) 9.00000i 0.341389i
\(696\) 0 0
\(697\) 0 0
\(698\) 36.0000 1.36262
\(699\) 10.0000 0.378235
\(700\) − 3.00000i − 0.113389i
\(701\) −4.00000 −0.151078 −0.0755390 0.997143i \(-0.524068\pi\)
−0.0755390 + 0.997143i \(0.524068\pi\)
\(702\) 0 0
\(703\) 5.00000 0.188579
\(704\) − 1.00000i − 0.0376889i
\(705\) 9.00000 0.338960
\(706\) 36.0000 1.35488
\(707\) 12.0000i 0.451306i
\(708\) 4.00000i 0.150329i
\(709\) 4.00000i 0.150223i 0.997175 + 0.0751116i \(0.0239313\pi\)
−0.997175 + 0.0751116i \(0.976069\pi\)
\(710\) − 2.00000i − 0.0750587i
\(711\) −10.0000 −0.375029
\(712\) −1.00000 −0.0374766
\(713\) − 40.0000i − 1.49801i
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) − 2.00000i − 0.0746914i
\(718\) 34.0000 1.26887
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) − 27.0000i − 1.00553i
\(722\) 6.00000i 0.223297i
\(723\) − 15.0000i − 0.557856i
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) 10.0000i 0.371135i
\(727\) 37.0000 1.37225 0.686127 0.727482i \(-0.259309\pi\)
0.686127 + 0.727482i \(0.259309\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 16.0000i 0.592187i
\(731\) 0 0
\(732\) −2.00000 −0.0739221
\(733\) 5.00000i 0.184679i 0.995728 + 0.0923396i \(0.0294345\pi\)
−0.995728 + 0.0923396i \(0.970565\pi\)
\(734\) 0 0
\(735\) − 2.00000i − 0.0737711i
\(736\) − 4.00000i − 0.147442i
\(737\) −12.0000 −0.442026
\(738\) −6.00000 −0.220863
\(739\) 35.0000i 1.28750i 0.765238 + 0.643748i \(0.222621\pi\)
−0.765238 + 0.643748i \(0.777379\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) 39.0000 1.43174
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) −10.0000 −0.366618
\(745\) 6.00000 0.219823
\(746\) − 10.0000i − 0.366126i
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) 18.0000i 0.657706i
\(750\) −1.00000 −0.0365148
\(751\) −44.0000 −1.60558 −0.802791 0.596260i \(-0.796653\pi\)
−0.802791 + 0.596260i \(0.796653\pi\)
\(752\) − 9.00000i − 0.328196i
\(753\) 11.0000 0.400862
\(754\) 0 0
\(755\) −6.00000 −0.218362
\(756\) − 3.00000i − 0.109109i
\(757\) 39.0000 1.41748 0.708740 0.705470i \(-0.249264\pi\)
0.708740 + 0.705470i \(0.249264\pi\)
\(758\) 1.00000 0.0363216
\(759\) − 4.00000i − 0.145191i
\(760\) 5.00000i 0.181369i
\(761\) 45.0000i 1.63125i 0.578582 + 0.815624i \(0.303606\pi\)
−0.578582 + 0.815624i \(0.696394\pi\)
\(762\) − 5.00000i − 0.181131i
\(763\) 30.0000 1.08607
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 28.0000 1.01168
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 26.0000i 0.937584i 0.883309 + 0.468792i \(0.155311\pi\)
−0.883309 + 0.468792i \(0.844689\pi\)
\(770\) −3.00000 −0.108112
\(771\) −24.0000 −0.864339
\(772\) − 16.0000i − 0.575853i
\(773\) 37.0000i 1.33080i 0.746488 + 0.665399i \(0.231738\pi\)
−0.746488 + 0.665399i \(0.768262\pi\)
\(774\) − 2.00000i − 0.0718885i
\(775\) 10.0000i 0.359211i
\(776\) 12.0000 0.430775
\(777\) 3.00000 0.107624
\(778\) − 16.0000i − 0.573628i
\(779\) 30.0000 1.07486
\(780\) 0 0
\(781\) −2.00000 −0.0715656
\(782\) 0 0
\(783\) 0 0
\(784\) −2.00000 −0.0714286
\(785\) − 1.00000i − 0.0356915i
\(786\) − 15.0000i − 0.535032i
\(787\) 16.0000i 0.570338i 0.958477 + 0.285169i \(0.0920498\pi\)
−0.958477 + 0.285169i \(0.907950\pi\)
\(788\) − 15.0000i − 0.534353i
\(789\) −3.00000 −0.106803
\(790\) 10.0000 0.355784
\(791\) − 48.0000i − 1.70668i
\(792\) −1.00000 −0.0355335
\(793\) 0 0
\(794\) −33.0000 −1.17113
\(795\) − 13.0000i − 0.461062i
\(796\) −2.00000 −0.0708881
\(797\) 14.0000 0.495905 0.247953 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(798\) 15.0000i 0.530994i
\(799\) 0 0
\(800\) 1.00000i 0.0353553i
\(801\) 1.00000i 0.0353333i
\(802\) 25.0000 0.882781
\(803\) 16.0000 0.564628
\(804\) 12.0000i 0.423207i
\(805\) −12.0000 −0.422944
\(806\) 0 0
\(807\) 20.0000 0.704033
\(808\) − 4.00000i − 0.140720i
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) −1.00000 −0.0351364
\(811\) − 33.0000i − 1.15879i −0.815048 0.579393i \(-0.803290\pi\)
0.815048 0.579393i \(-0.196710\pi\)
\(812\) 0 0
\(813\) 24.0000i 0.841717i
\(814\) − 1.00000i − 0.0350500i
\(815\) −20.0000 −0.700569
\(816\) 0 0
\(817\) 10.0000i 0.349856i
\(818\) −17.0000 −0.594391
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) − 18.0000i − 0.628204i −0.949389 0.314102i \(-0.898297\pi\)
0.949389 0.314102i \(-0.101703\pi\)
\(822\) −16.0000 −0.558064
\(823\) 5.00000 0.174289 0.0871445 0.996196i \(-0.472226\pi\)
0.0871445 + 0.996196i \(0.472226\pi\)
\(824\) 9.00000i 0.313530i
\(825\) 1.00000i 0.0348155i
\(826\) − 12.0000i − 0.417533i
\(827\) − 30.0000i − 1.04320i −0.853189 0.521601i \(-0.825335\pi\)
0.853189 0.521601i \(-0.174665\pi\)
\(828\) −4.00000 −0.139010
\(829\) 44.0000 1.52818 0.764092 0.645108i \(-0.223188\pi\)
0.764092 + 0.645108i \(0.223188\pi\)
\(830\) 12.0000i 0.416526i
\(831\) 23.0000 0.797861
\(832\) 0 0
\(833\) 0 0
\(834\) − 9.00000i − 0.311645i
\(835\) −13.0000 −0.449884
\(836\) 5.00000 0.172929
\(837\) 10.0000i 0.345651i
\(838\) 28.0000i 0.967244i
\(839\) − 54.0000i − 1.86429i −0.362089 0.932144i \(-0.617936\pi\)
0.362089 0.932144i \(-0.382064\pi\)
\(840\) 3.00000i 0.103510i
\(841\) −29.0000 −1.00000
\(842\) −20.0000 −0.689246
\(843\) − 10.0000i − 0.344418i
\(844\) 15.0000 0.516321
\(845\) 0 0
\(846\) −9.00000 −0.309426
\(847\) − 30.0000i − 1.03081i
\(848\) −13.0000 −0.446422
\(849\) 2.00000 0.0686398
\(850\) 0 0
\(851\) − 4.00000i − 0.137118i
\(852\) 2.00000i 0.0685189i
\(853\) − 14.0000i − 0.479351i −0.970853 0.239675i \(-0.922959\pi\)
0.970853 0.239675i \(-0.0770410\pi\)
\(854\) 6.00000 0.205316
\(855\) 5.00000 0.170996
\(856\) − 6.00000i − 0.205076i
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 19.0000 0.648272 0.324136 0.946011i \(-0.394927\pi\)
0.324136 + 0.946011i \(0.394927\pi\)
\(860\) 2.00000i 0.0681994i
\(861\) 18.0000 0.613438
\(862\) −36.0000 −1.22616
\(863\) 48.0000i 1.63394i 0.576681 + 0.816970i \(0.304348\pi\)
−0.576681 + 0.816970i \(0.695652\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) − 9.00000i − 0.306009i
\(866\) 16.0000i 0.543702i
\(867\) 17.0000 0.577350
\(868\) 30.0000 1.01827
\(869\) − 10.0000i − 0.339227i
\(870\) 0 0
\(871\) 0 0
\(872\) −10.0000 −0.338643
\(873\) − 12.0000i − 0.406138i
\(874\) 20.0000 0.676510
\(875\) 3.00000 0.101419
\(876\) − 16.0000i − 0.540590i
\(877\) − 38.0000i − 1.28317i −0.767052 0.641584i \(-0.778277\pi\)
0.767052 0.641584i \(-0.221723\pi\)
\(878\) − 10.0000i − 0.337484i
\(879\) − 13.0000i − 0.438479i
\(880\) 1.00000 0.0337100
\(881\) −5.00000 −0.168454 −0.0842271 0.996447i \(-0.526842\pi\)
−0.0842271 + 0.996447i \(0.526842\pi\)
\(882\) 2.00000i 0.0673435i
\(883\) −42.0000 −1.41341 −0.706706 0.707507i \(-0.749820\pi\)
−0.706706 + 0.707507i \(0.749820\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) 18.0000i 0.604722i
\(887\) −13.0000 −0.436497 −0.218249 0.975893i \(-0.570034\pi\)
−0.218249 + 0.975893i \(0.570034\pi\)
\(888\) −1.00000 −0.0335578
\(889\) 15.0000i 0.503084i
\(890\) − 1.00000i − 0.0335201i
\(891\) 1.00000i 0.0335013i
\(892\) 11.0000i 0.368307i
\(893\) 45.0000 1.50587
\(894\) −6.00000 −0.200670
\(895\) 12.0000i 0.401116i
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) −15.0000 −0.500556
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) − 6.00000i − 0.199778i
\(903\) 6.00000i 0.199667i
\(904\) 16.0000i 0.532152i
\(905\) 6.00000i 0.199447i
\(906\) 6.00000 0.199337
\(907\) −42.0000 −1.39459 −0.697294 0.716786i \(-0.745613\pi\)
−0.697294 + 0.716786i \(0.745613\pi\)
\(908\) − 20.0000i − 0.663723i
\(909\) −4.00000 −0.132672
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) − 5.00000i − 0.165567i
\(913\) 12.0000 0.397142
\(914\) −22.0000 −0.727695
\(915\) − 2.00000i − 0.0661180i
\(916\) 10.0000i 0.330409i
\(917\) 45.0000i 1.48603i
\(918\) 0 0
\(919\) 34.0000 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) 4.00000 0.131876
\(921\) − 18.0000i − 0.593120i
\(922\) 12.0000 0.395199
\(923\) 0 0
\(924\) 3.00000 0.0986928
\(925\) 1.00000i 0.0328798i
\(926\) −16.0000 −0.525793
\(927\) 9.00000 0.295599
\(928\) 0 0
\(929\) − 34.0000i − 1.11550i −0.830008 0.557752i \(-0.811664\pi\)
0.830008 0.557752i \(-0.188336\pi\)
\(930\) − 10.0000i − 0.327913i
\(931\) − 10.0000i − 0.327737i
\(932\) 10.0000 0.327561
\(933\) −12.0000 −0.392862
\(934\) − 36.0000i − 1.17796i
\(935\) 0 0
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) − 36.0000i − 1.17544i
\(939\) 34.0000 1.10955
\(940\) 9.00000 0.293548
\(941\) − 24.0000i − 0.782378i −0.920310 0.391189i \(-0.872064\pi\)
0.920310 0.391189i \(-0.127936\pi\)
\(942\) 1.00000i 0.0325818i
\(943\) − 24.0000i − 0.781548i
\(944\) 4.00000i 0.130189i
\(945\) 3.00000 0.0975900
\(946\) 2.00000 0.0650256
\(947\) − 38.0000i − 1.23483i −0.786636 0.617417i \(-0.788179\pi\)
0.786636 0.617417i \(-0.211821\pi\)
\(948\) −10.0000 −0.324785
\(949\) 0 0
\(950\) −5.00000 −0.162221
\(951\) 19.0000i 0.616117i
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 13.0000i 0.420891i
\(955\) 18.0000i 0.582466i
\(956\) − 2.00000i − 0.0646846i
\(957\) 0 0
\(958\) −8.00000 −0.258468
\(959\) 48.0000 1.55000
\(960\) − 1.00000i − 0.0322749i
\(961\) −69.0000 −2.22581
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) − 15.0000i − 0.483117i
\(965\) 16.0000 0.515058
\(966\) 12.0000 0.386094
\(967\) − 7.00000i − 0.225105i −0.993646 0.112552i \(-0.964097\pi\)
0.993646 0.112552i \(-0.0359026\pi\)
\(968\) 10.0000i 0.321412i
\(969\) 0 0
\(970\) 12.0000i 0.385297i
\(971\) 27.0000 0.866471 0.433236 0.901281i \(-0.357372\pi\)
0.433236 + 0.901281i \(0.357372\pi\)
\(972\) 1.00000 0.0320750
\(973\) 27.0000i 0.865580i
\(974\) −35.0000 −1.12147
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) − 48.0000i − 1.53566i −0.640656 0.767828i \(-0.721338\pi\)
0.640656 0.767828i \(-0.278662\pi\)
\(978\) 20.0000 0.639529
\(979\) −1.00000 −0.0319601
\(980\) − 2.00000i − 0.0638877i
\(981\) 10.0000i 0.319275i
\(982\) 25.0000i 0.797782i
\(983\) 53.0000i 1.69044i 0.534421 + 0.845219i \(0.320530\pi\)
−0.534421 + 0.845219i \(0.679470\pi\)
\(984\) −6.00000 −0.191273
\(985\) 15.0000 0.477940
\(986\) 0 0
\(987\) 27.0000 0.859419
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) − 1.00000i − 0.0317821i
\(991\) 38.0000 1.20711 0.603555 0.797321i \(-0.293750\pi\)
0.603555 + 0.797321i \(0.293750\pi\)
\(992\) −10.0000 −0.317500
\(993\) 28.0000i 0.888553i
\(994\) − 6.00000i − 0.190308i
\(995\) − 2.00000i − 0.0634043i
\(996\) − 12.0000i − 0.380235i
\(997\) −7.00000 −0.221692 −0.110846 0.993838i \(-0.535356\pi\)
−0.110846 + 0.993838i \(0.535356\pi\)
\(998\) −20.0000 −0.633089
\(999\) 1.00000i 0.0316386i
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 5070.2.b.a.1351.1 2
13.5 odd 4 5070.2.a.c.1.1 1
13.7 odd 12 390.2.i.b.211.1 yes 2
13.8 odd 4 5070.2.a.q.1.1 1
13.11 odd 12 390.2.i.b.61.1 2
13.12 even 2 inner 5070.2.b.a.1351.2 2
39.11 even 12 1170.2.i.j.451.1 2
39.20 even 12 1170.2.i.j.991.1 2
65.7 even 12 1950.2.z.i.1849.2 4
65.24 odd 12 1950.2.i.o.451.1 2
65.33 even 12 1950.2.z.i.1849.1 4
65.37 even 12 1950.2.z.i.1699.1 4
65.59 odd 12 1950.2.i.o.601.1 2
65.63 even 12 1950.2.z.i.1699.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.b.61.1 2 13.11 odd 12
390.2.i.b.211.1 yes 2 13.7 odd 12
1170.2.i.j.451.1 2 39.11 even 12
1170.2.i.j.991.1 2 39.20 even 12
1950.2.i.o.451.1 2 65.24 odd 12
1950.2.i.o.601.1 2 65.59 odd 12
1950.2.z.i.1699.1 4 65.37 even 12
1950.2.z.i.1699.2 4 65.63 even 12
1950.2.z.i.1849.1 4 65.33 even 12
1950.2.z.i.1849.2 4 65.7 even 12
5070.2.a.c.1.1 1 13.5 odd 4
5070.2.a.q.1.1 1 13.8 odd 4
5070.2.b.a.1351.1 2 1.1 even 1 trivial
5070.2.b.a.1351.2 2 13.12 even 2 inner