Properties

Label 5070.2.b.a.1351.2
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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

$q$-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 \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) − 1.00000i − 0.301511i −0.988571 0.150756i \(-0.951829\pi\)
0.988571 0.150756i \(-0.0481707\pi\)
\(12\) 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 \(-0.805570\pi\)
0.819178 0.573539i \(-0.194430\pi\)
\(20\) − 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 \(0.354999\pi\)
−0.439941 + 0.898027i \(0.645001\pi\)
\(32\) 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 \(0.0261945\pi\)
−0.996616 + 0.0821995i \(0.973806\pi\)
\(38\) 5.00000 0.811107
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 6.00000i 0.937043i 0.883452 + 0.468521i \(0.155213\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) 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 \(0.227918\pi\)
−0.754420 + 0.656392i \(0.772082\pi\)
\(48\) −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 \(-0.916153\pi\)
0.965507 0.260378i \(-0.0838471\pi\)
\(60\) 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 \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) − 3.00000i − 0.358569i
\(71\) − 2.00000i − 0.237356i −0.992933 0.118678i \(-0.962134\pi\)
0.992933 0.118678i \(-0.0378657\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 16.0000i 1.87266i 0.351123 + 0.936329i \(0.385800\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −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 \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 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 \(-0.983122\pi\)
0.998595 0.0529999i \(-0.0168783\pi\)
\(90\) −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 \(0.208512\pi\)
−0.793011 + 0.609208i \(0.791488\pi\)
\(98\) − 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 \(-0.841031\pi\)
0.877862 0.478913i \(-0.158969\pi\)
\(110\) 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 \(-0.760463\pi\)
0.729964 0.683486i \(-0.239537\pi\)
\(138\) − 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 \(-0.920959\pi\)
0.969328 0.245770i \(-0.0790407\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 6.00000i 0.488273i 0.969741 + 0.244137i \(0.0785045\pi\)
−0.969741 + 0.244137i \(0.921495\pi\)
\(152\) −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 \(0.286445\pi\)
−0.621694 + 0.783260i \(0.713555\pi\)
\(164\) − 6.00000i − 0.468521i
\(165\) −1.00000 −0.0778499
\(166\) −12.0000 −0.931381
\(167\) 13.0000i 1.00597i 0.864295 + 0.502985i \(0.167765\pi\)
−0.864295 + 0.502985i \(0.832235\pi\)
\(168\) −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 \(-0.804670\pi\)
0.817554 0.575853i \(-0.195330\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) − 15.0000i − 1.06871i −0.845262 0.534353i \(-0.820555\pi\)
0.845262 0.534353i \(-0.179445\pi\)
\(198\) 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 \(0.120063\pi\)
−0.929704 + 0.368307i \(0.879937\pi\)
\(224\) −3.00000 −0.200446
\(225\) −1.00000 −0.0666667
\(226\) 16.0000i 1.06430i
\(227\) − 20.0000i − 1.32745i −0.747978 0.663723i \(-0.768975\pi\)
0.747978 0.663723i \(-0.231025\pi\)
\(228\) − 5.00000i − 0.331133i
\(229\) 10.0000i 0.660819i 0.943838 + 0.330409i \(0.107187\pi\)
−0.943838 + 0.330409i \(0.892813\pi\)
\(230\) −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 \(-0.979396\pi\)
0.997906 0.0646846i \(-0.0206041\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) − 15.0000i − 0.966235i −0.875556 0.483117i \(-0.839504\pi\)
0.875556 0.483117i \(-0.160496\pi\)
\(242\) 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 \(0.259990\pi\)
−0.684569 + 0.728948i \(0.740010\pi\)
\(272\) 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 \(-0.903589\pi\)
0.954480 0.298275i \(-0.0964112\pi\)
\(282\) 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 \(-0.876015\pi\)
0.925096 0.379734i \(-0.123985\pi\)
\(294\) 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 \(-0.828290\pi\)
0.857996 0.513657i \(-0.171710\pi\)
\(308\) −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 \(0.179151\pi\)
−0.845754 + 0.533573i \(0.820849\pi\)
\(318\) 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 \(0.279499\pi\)
−0.638635 + 0.769510i \(0.720501\pi\)
\(332\) − 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 \(-0.586245\pi\)
0.267644 0.963518i \(-0.413755\pi\)
\(350\) 3.00000 0.160357
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) − 36.0000i − 1.91609i −0.286623 0.958043i \(-0.592533\pi\)
0.286623 0.958043i \(-0.407467\pi\)
\(354\) −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 \(-0.645579\pi\)
0.441572 0.897226i \(-0.354421\pi\)
\(360\) 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 \(-0.991824\pi\)
0.999670 0.0256833i \(-0.00817614\pi\)
\(380\) −5.00000 −0.256495
\(381\) 5.00000 0.256158
\(382\) − 18.0000i − 0.920960i
\(383\) − 28.0000i − 1.43073i −0.698749 0.715367i \(-0.746260\pi\)
0.698749 0.715367i \(-0.253740\pi\)
\(384\) 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 \(0.310584\pi\)
−0.560564 + 0.828111i \(0.689416\pi\)
\(398\) 2.00000i 0.100251i
\(399\) −15.0000 −0.750939
\(400\) −1.00000 −0.0500000
\(401\) − 25.0000i − 1.24844i −0.781248 0.624220i \(-0.785417\pi\)
0.781248 0.624220i \(-0.214583\pi\)
\(402\) −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 \(0.138074\pi\)
−0.907386 + 0.420298i \(0.861926\pi\)
\(410\) − 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 \(0.162044\pi\)
−0.873195 + 0.487370i \(0.837956\pi\)
\(422\) − 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 \(0.333974\pi\)
−0.498257 + 0.867029i \(0.666026\pi\)
\(432\) −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 \(0.115161\pi\)
−0.935266 + 0.353947i \(0.884839\pi\)
\(450\) − 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 \(0.172044\pi\)
−0.857455 + 0.514558i \(0.827956\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) − 4.00000i − 0.186501i
\(461\) − 12.0000i − 0.558896i −0.960161 0.279448i \(-0.909849\pi\)
0.960161 0.279448i \(-0.0901514\pi\)
\(462\) − 3.00000i − 0.139573i
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) 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 \(0.0585046\pi\)
−0.983157 + 0.182765i \(0.941495\pi\)
\(480\) 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 \(0.291482\pi\)
−0.609221 + 0.793001i \(0.708518\pi\)
\(488\) 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 \(0.147743\pi\)
−0.894203 + 0.447661i \(0.852257\pi\)
\(500\) 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 \(0.130614\pi\)
−0.916987 + 0.398918i \(0.869386\pi\)
\(510\) 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 \(0.0136894\pi\)
−0.999075 + 0.0429934i \(0.986311\pi\)
\(542\) −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 \(0.0202445\pi\)
−0.997978 + 0.0635570i \(0.979756\pi\)
\(558\) −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 \(0.122246\pi\)
−0.927156 + 0.374675i \(0.877754\pi\)
\(578\) − 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 \(-0.878854\pi\)
0.928445 0.371470i \(-0.121146\pi\)
\(588\) −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 \(-0.805037\pi\)
0.818216 0.574911i \(-0.194963\pi\)
\(594\) −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 \(0.153759\pi\)
−0.885583 + 0.464481i \(0.846241\pi\)
\(614\) 18.0000 0.726421
\(615\) 6.00000 0.241943
\(616\) − 3.00000i − 0.120873i
\(617\) − 6.00000i − 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) − 9.00000i − 0.362033i
\(619\) 31.0000i 1.24600i 0.782224 + 0.622998i \(0.214085\pi\)
−0.782224 + 0.622998i \(0.785915\pi\)
\(620\) 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 \(0.0253703\pi\)
−0.996825 + 0.0796187i \(0.974630\pi\)
\(632\) 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 \(-0.665665\pi\)
0.497271 0.867595i \(-0.334335\pi\)
\(644\) − 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 \(-0.912226\pi\)
0.962221 0.272268i \(-0.0877739\pi\)
\(662\) −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 \(0.165723\pi\)
−0.867503 + 0.497431i \(0.834277\pi\)
\(684\) 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 \(0.215999\pi\)
−0.778464 + 0.627690i \(0.784001\pi\)
\(692\) −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 \(-0.976069\pi\)
0.997175 0.0751116i \(-0.0239313\pi\)
\(710\) 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 \(-0.970565\pi\)
0.995728 0.0923396i \(-0.0294345\pi\)
\(734\) 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 \(-0.777379\pi\)
0.765238 0.643748i \(-0.222621\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) 39.0000 1.43174
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) −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 \(-0.696394\pi\)
0.578582 0.815624i \(-0.303606\pi\)
\(762\) 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 \(-0.844689\pi\)
0.883309 0.468792i \(-0.155311\pi\)
\(770\) −3.00000 −0.108112
\(771\) −24.0000 −0.864339
\(772\) 16.0000i 0.575853i
\(773\) − 37.0000i − 1.33080i −0.746488 0.665399i \(-0.768262\pi\)
0.746488 0.665399i \(-0.231738\pi\)
\(774\) 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 \(-0.907950\pi\)
0.958477 0.285169i \(-0.0920498\pi\)
\(788\) 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 \(0.196710\pi\)
−0.815048 + 0.579393i \(0.803290\pi\)
\(812\) 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 \(0.101703\pi\)
−0.949389 + 0.314102i \(0.898297\pi\)
\(822\) −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 \(0.174665\pi\)
−0.853189 + 0.521601i \(0.825335\pi\)
\(828\) −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 \(0.382064\pi\)
−0.362089 + 0.932144i \(0.617936\pi\)
\(840\) − 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 \(0.0770410\pi\)
−0.970853 + 0.239675i \(0.922959\pi\)
\(854\) 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 \(-0.695652\pi\)
0.576681 0.816970i \(-0.304348\pi\)
\(864\) − 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 \(0.221723\pi\)
−0.767052 + 0.641584i \(0.778277\pi\)
\(878\) 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 \(0.188336\pi\)
−0.830008 + 0.557752i \(0.811664\pi\)
\(930\) 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 \(0.127936\pi\)
−0.920310 + 0.391189i \(0.872064\pi\)
\(942\) − 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 \(0.211821\pi\)
−0.786636 + 0.617417i \(0.788179\pi\)
\(948\) −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 \(0.0359026\pi\)
−0.993646 + 0.112552i \(0.964097\pi\)
\(968\) − 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 \(0.278662\pi\)
−0.640656 + 0.767828i \(0.721338\pi\)
\(978\) 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 \(-0.679470\pi\)
0.534421 0.845219i \(-0.320530\pi\)
\(984\) −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.1351.2 2
13.2 odd 12 390.2.i.b.61.1 2
13.5 odd 4 5070.2.a.q.1.1 1
13.6 odd 12 390.2.i.b.211.1 yes 2
13.8 odd 4 5070.2.a.c.1.1 1
13.12 even 2 inner 5070.2.b.a.1351.1 2
39.2 even 12 1170.2.i.j.451.1 2
39.32 even 12 1170.2.i.j.991.1 2
65.2 even 12 1950.2.z.i.1699.1 4
65.19 odd 12 1950.2.i.o.601.1 2
65.28 even 12 1950.2.z.i.1699.2 4
65.32 even 12 1950.2.z.i.1849.2 4
65.54 odd 12 1950.2.i.o.451.1 2
65.58 even 12 1950.2.z.i.1849.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.b.61.1 2 13.2 odd 12
390.2.i.b.211.1 yes 2 13.6 odd 12
1170.2.i.j.451.1 2 39.2 even 12
1170.2.i.j.991.1 2 39.32 even 12
1950.2.i.o.451.1 2 65.54 odd 12
1950.2.i.o.601.1 2 65.19 odd 12
1950.2.z.i.1699.1 4 65.2 even 12
1950.2.z.i.1699.2 4 65.28 even 12
1950.2.z.i.1849.1 4 65.58 even 12
1950.2.z.i.1849.2 4 65.32 even 12
5070.2.a.c.1.1 1 13.8 odd 4
5070.2.a.q.1.1 1 13.5 odd 4
5070.2.b.a.1351.1 2 13.12 even 2 inner
5070.2.b.a.1351.2 2 1.1 even 1 trivial