Properties

Label 5070.2.b.r.1351.2
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.r.1351.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +3.56155i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +3.56155i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.12311i q^{11} -1.00000 q^{12} +3.56155 q^{14} -1.00000i q^{15} +1.00000 q^{16} -5.12311 q^{17} -1.00000i q^{18} +3.56155i q^{19} +1.00000i q^{20} +3.56155i q^{21} -4.12311 q^{22} +7.68466 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -3.56155i q^{28} +6.56155 q^{29} -1.00000 q^{30} -5.68466i q^{31} -1.00000i q^{32} -4.12311i q^{33} +5.12311i q^{34} +3.56155 q^{35} -1.00000 q^{36} +4.12311i q^{37} +3.56155 q^{38} +1.00000 q^{40} +4.24621i q^{41} +3.56155 q^{42} -4.56155 q^{43} +4.12311i q^{44} -1.00000i q^{45} -7.68466i q^{46} +7.00000i q^{47} +1.00000 q^{48} -5.68466 q^{49} +1.00000i q^{50} -5.12311 q^{51} +4.43845 q^{53} -1.00000i q^{54} -4.12311 q^{55} -3.56155 q^{56} +3.56155i q^{57} -6.56155i q^{58} +10.5616i q^{59} +1.00000i q^{60} +6.00000 q^{61} -5.68466 q^{62} +3.56155i q^{63} -1.00000 q^{64} -4.12311 q^{66} -14.2462i q^{67} +5.12311 q^{68} +7.68466 q^{69} -3.56155i q^{70} +4.87689i q^{71} +1.00000i q^{72} -15.3693i q^{73} +4.12311 q^{74} -1.00000 q^{75} -3.56155i q^{76} +14.6847 q^{77} +7.43845 q^{79} -1.00000i q^{80} +1.00000 q^{81} +4.24621 q^{82} -1.12311i q^{83} -3.56155i q^{84} +5.12311i q^{85} +4.56155i q^{86} +6.56155 q^{87} +4.12311 q^{88} +1.80776i q^{89} -1.00000 q^{90} -7.68466 q^{92} -5.68466i q^{93} +7.00000 q^{94} +3.56155 q^{95} -1.00000i q^{96} -1.12311i q^{97} +5.68466i q^{98} -4.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 4q^{4} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 4q^{4} + 4q^{9} - 4q^{10} - 4q^{12} + 6q^{14} + 4q^{16} - 4q^{17} + 6q^{23} - 4q^{25} + 4q^{27} + 18q^{29} - 4q^{30} + 6q^{35} - 4q^{36} + 6q^{38} + 4q^{40} + 6q^{42} - 10q^{43} + 4q^{48} + 2q^{49} - 4q^{51} + 26q^{53} - 6q^{56} + 24q^{61} + 2q^{62} - 4q^{64} + 4q^{68} + 6q^{69} - 4q^{75} + 34q^{77} + 38q^{79} + 4q^{81} - 16q^{82} + 18q^{87} - 4q^{90} - 6q^{92} + 28q^{94} + 6q^{95} + O(q^{100}) \)

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
\(7\) 3.56155i 1.34614i 0.739579 + 0.673070i \(0.235025\pi\)
−0.739579 + 0.673070i \(0.764975\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) − 4.12311i − 1.24316i −0.783349 0.621582i \(-0.786490\pi\)
0.783349 0.621582i \(-0.213510\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 3.56155 0.951865
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) −5.12311 −1.24254 −0.621268 0.783598i \(-0.713382\pi\)
−0.621268 + 0.783598i \(0.713382\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 3.56155i 0.817076i 0.912741 + 0.408538i \(0.133961\pi\)
−0.912741 + 0.408538i \(0.866039\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 3.56155i 0.777195i
\(22\) −4.12311 −0.879049
\(23\) 7.68466 1.60236 0.801181 0.598422i \(-0.204205\pi\)
0.801181 + 0.598422i \(0.204205\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) − 3.56155i − 0.673070i
\(29\) 6.56155 1.21845 0.609225 0.792998i \(-0.291481\pi\)
0.609225 + 0.792998i \(0.291481\pi\)
\(30\) −1.00000 −0.182574
\(31\) − 5.68466i − 1.02099i −0.859879 0.510497i \(-0.829461\pi\)
0.859879 0.510497i \(-0.170539\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 4.12311i − 0.717741i
\(34\) 5.12311i 0.878605i
\(35\) 3.56155 0.602012
\(36\) −1.00000 −0.166667
\(37\) 4.12311i 0.677834i 0.940816 + 0.338917i \(0.110061\pi\)
−0.940816 + 0.338917i \(0.889939\pi\)
\(38\) 3.56155 0.577760
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 4.24621i 0.663147i 0.943429 + 0.331573i \(0.107579\pi\)
−0.943429 + 0.331573i \(0.892421\pi\)
\(42\) 3.56155 0.549560
\(43\) −4.56155 −0.695630 −0.347815 0.937563i \(-0.613076\pi\)
−0.347815 + 0.937563i \(0.613076\pi\)
\(44\) 4.12311i 0.621582i
\(45\) − 1.00000i − 0.149071i
\(46\) − 7.68466i − 1.13304i
\(47\) 7.00000i 1.02105i 0.859861 + 0.510527i \(0.170550\pi\)
−0.859861 + 0.510527i \(0.829450\pi\)
\(48\) 1.00000 0.144338
\(49\) −5.68466 −0.812094
\(50\) 1.00000i 0.141421i
\(51\) −5.12311 −0.717378
\(52\) 0 0
\(53\) 4.43845 0.609668 0.304834 0.952406i \(-0.401399\pi\)
0.304834 + 0.952406i \(0.401399\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −4.12311 −0.555959
\(56\) −3.56155 −0.475933
\(57\) 3.56155i 0.471739i
\(58\) − 6.56155i − 0.861574i
\(59\) 10.5616i 1.37500i 0.726186 + 0.687499i \(0.241291\pi\)
−0.726186 + 0.687499i \(0.758709\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −5.68466 −0.721952
\(63\) 3.56155i 0.448713i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.12311 −0.507519
\(67\) − 14.2462i − 1.74045i −0.492653 0.870226i \(-0.663973\pi\)
0.492653 0.870226i \(-0.336027\pi\)
\(68\) 5.12311 0.621268
\(69\) 7.68466 0.925124
\(70\) − 3.56155i − 0.425687i
\(71\) 4.87689i 0.578781i 0.957211 + 0.289390i \(0.0934526\pi\)
−0.957211 + 0.289390i \(0.906547\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 15.3693i − 1.79884i −0.437083 0.899421i \(-0.643988\pi\)
0.437083 0.899421i \(-0.356012\pi\)
\(74\) 4.12311 0.479301
\(75\) −1.00000 −0.115470
\(76\) − 3.56155i − 0.408538i
\(77\) 14.6847 1.67347
\(78\) 0 0
\(79\) 7.43845 0.836891 0.418445 0.908242i \(-0.362575\pi\)
0.418445 + 0.908242i \(0.362575\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) 4.24621 0.468916
\(83\) − 1.12311i − 0.123277i −0.998099 0.0616384i \(-0.980367\pi\)
0.998099 0.0616384i \(-0.0196326\pi\)
\(84\) − 3.56155i − 0.388597i
\(85\) 5.12311i 0.555679i
\(86\) 4.56155i 0.491885i
\(87\) 6.56155 0.703472
\(88\) 4.12311 0.439525
\(89\) 1.80776i 0.191623i 0.995400 + 0.0958113i \(0.0305445\pi\)
−0.995400 + 0.0958113i \(0.969455\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −7.68466 −0.801181
\(93\) − 5.68466i − 0.589472i
\(94\) 7.00000 0.721995
\(95\) 3.56155 0.365408
\(96\) − 1.00000i − 0.102062i
\(97\) − 1.12311i − 0.114034i −0.998373 0.0570170i \(-0.981841\pi\)
0.998373 0.0570170i \(-0.0181589\pi\)
\(98\) 5.68466i 0.574237i
\(99\) − 4.12311i − 0.414388i
\(100\) 1.00000 0.100000
\(101\) 17.1231 1.70381 0.851906 0.523694i \(-0.175447\pi\)
0.851906 + 0.523694i \(0.175447\pi\)
\(102\) 5.12311i 0.507263i
\(103\) −0.438447 −0.0432015 −0.0216007 0.999767i \(-0.506876\pi\)
−0.0216007 + 0.999767i \(0.506876\pi\)
\(104\) 0 0
\(105\) 3.56155 0.347572
\(106\) − 4.43845i − 0.431100i
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 20.2462i 1.93924i 0.244625 + 0.969618i \(0.421335\pi\)
−0.244625 + 0.969618i \(0.578665\pi\)
\(110\) 4.12311i 0.393123i
\(111\) 4.12311i 0.391348i
\(112\) 3.56155i 0.336535i
\(113\) −3.68466 −0.346624 −0.173312 0.984867i \(-0.555447\pi\)
−0.173312 + 0.984867i \(0.555447\pi\)
\(114\) 3.56155 0.333570
\(115\) − 7.68466i − 0.716598i
\(116\) −6.56155 −0.609225
\(117\) 0 0
\(118\) 10.5616 0.972270
\(119\) − 18.2462i − 1.67263i
\(120\) 1.00000 0.0912871
\(121\) −6.00000 −0.545455
\(122\) − 6.00000i − 0.543214i
\(123\) 4.24621i 0.382868i
\(124\) 5.68466i 0.510497i
\(125\) 1.00000i 0.0894427i
\(126\) 3.56155 0.317288
\(127\) 4.43845 0.393849 0.196924 0.980419i \(-0.436905\pi\)
0.196924 + 0.980419i \(0.436905\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −4.56155 −0.401622
\(130\) 0 0
\(131\) 6.12311 0.534978 0.267489 0.963561i \(-0.413806\pi\)
0.267489 + 0.963561i \(0.413806\pi\)
\(132\) 4.12311i 0.358870i
\(133\) −12.6847 −1.09990
\(134\) −14.2462 −1.23069
\(135\) − 1.00000i − 0.0860663i
\(136\) − 5.12311i − 0.439303i
\(137\) − 8.80776i − 0.752498i −0.926519 0.376249i \(-0.877214\pi\)
0.926519 0.376249i \(-0.122786\pi\)
\(138\) − 7.68466i − 0.654162i
\(139\) 8.43845 0.715740 0.357870 0.933771i \(-0.383503\pi\)
0.357870 + 0.933771i \(0.383503\pi\)
\(140\) −3.56155 −0.301006
\(141\) 7.00000i 0.589506i
\(142\) 4.87689 0.409260
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 6.56155i − 0.544907i
\(146\) −15.3693 −1.27197
\(147\) −5.68466 −0.468863
\(148\) − 4.12311i − 0.338917i
\(149\) 22.8078i 1.86848i 0.356639 + 0.934242i \(0.383923\pi\)
−0.356639 + 0.934242i \(0.616077\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 9.36932i 0.762464i 0.924479 + 0.381232i \(0.124500\pi\)
−0.924479 + 0.381232i \(0.875500\pi\)
\(152\) −3.56155 −0.288880
\(153\) −5.12311 −0.414179
\(154\) − 14.6847i − 1.18332i
\(155\) −5.68466 −0.456603
\(156\) 0 0
\(157\) 22.1231 1.76562 0.882808 0.469734i \(-0.155650\pi\)
0.882808 + 0.469734i \(0.155650\pi\)
\(158\) − 7.43845i − 0.591771i
\(159\) 4.43845 0.351992
\(160\) −1.00000 −0.0790569
\(161\) 27.3693i 2.15700i
\(162\) − 1.00000i − 0.0785674i
\(163\) − 18.5616i − 1.45385i −0.686715 0.726927i \(-0.740948\pi\)
0.686715 0.726927i \(-0.259052\pi\)
\(164\) − 4.24621i − 0.331573i
\(165\) −4.12311 −0.320983
\(166\) −1.12311 −0.0871699
\(167\) − 20.3693i − 1.57623i −0.615531 0.788113i \(-0.711058\pi\)
0.615531 0.788113i \(-0.288942\pi\)
\(168\) −3.56155 −0.274780
\(169\) 0 0
\(170\) 5.12311 0.392924
\(171\) 3.56155i 0.272359i
\(172\) 4.56155 0.347815
\(173\) 25.1771 1.91418 0.957089 0.289794i \(-0.0935868\pi\)
0.957089 + 0.289794i \(0.0935868\pi\)
\(174\) − 6.56155i − 0.497430i
\(175\) − 3.56155i − 0.269228i
\(176\) − 4.12311i − 0.310791i
\(177\) 10.5616i 0.793855i
\(178\) 1.80776 0.135498
\(179\) 0.315342 0.0235697 0.0117849 0.999931i \(-0.496249\pi\)
0.0117849 + 0.999931i \(0.496249\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) −11.1231 −0.826774 −0.413387 0.910555i \(-0.635654\pi\)
−0.413387 + 0.910555i \(0.635654\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 7.68466i 0.566521i
\(185\) 4.12311 0.303137
\(186\) −5.68466 −0.416819
\(187\) 21.1231i 1.54467i
\(188\) − 7.00000i − 0.510527i
\(189\) 3.56155i 0.259065i
\(190\) − 3.56155i − 0.258382i
\(191\) 19.1231 1.38370 0.691850 0.722042i \(-0.256796\pi\)
0.691850 + 0.722042i \(0.256796\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) −1.12311 −0.0806343
\(195\) 0 0
\(196\) 5.68466 0.406047
\(197\) 7.80776i 0.556280i 0.960541 + 0.278140i \(0.0897179\pi\)
−0.960541 + 0.278140i \(0.910282\pi\)
\(198\) −4.12311 −0.293016
\(199\) −11.1231 −0.788496 −0.394248 0.919004i \(-0.628995\pi\)
−0.394248 + 0.919004i \(0.628995\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) − 14.2462i − 1.00485i
\(202\) − 17.1231i − 1.20478i
\(203\) 23.3693i 1.64020i
\(204\) 5.12311 0.358689
\(205\) 4.24621 0.296568
\(206\) 0.438447i 0.0305481i
\(207\) 7.68466 0.534121
\(208\) 0 0
\(209\) 14.6847 1.01576
\(210\) − 3.56155i − 0.245770i
\(211\) 6.93087 0.477141 0.238570 0.971125i \(-0.423321\pi\)
0.238570 + 0.971125i \(0.423321\pi\)
\(212\) −4.43845 −0.304834
\(213\) 4.87689i 0.334159i
\(214\) − 2.00000i − 0.136717i
\(215\) 4.56155i 0.311095i
\(216\) 1.00000i 0.0680414i
\(217\) 20.2462 1.37440
\(218\) 20.2462 1.37125
\(219\) − 15.3693i − 1.03856i
\(220\) 4.12311 0.277980
\(221\) 0 0
\(222\) 4.12311 0.276725
\(223\) 26.3002i 1.76119i 0.473869 + 0.880595i \(0.342857\pi\)
−0.473869 + 0.880595i \(0.657143\pi\)
\(224\) 3.56155 0.237966
\(225\) −1.00000 −0.0666667
\(226\) 3.68466i 0.245100i
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) − 3.56155i − 0.235870i
\(229\) − 7.75379i − 0.512385i −0.966626 0.256192i \(-0.917532\pi\)
0.966626 0.256192i \(-0.0824681\pi\)
\(230\) −7.68466 −0.506711
\(231\) 14.6847 0.966180
\(232\) 6.56155i 0.430787i
\(233\) 17.6847 1.15856 0.579280 0.815128i \(-0.303334\pi\)
0.579280 + 0.815128i \(0.303334\pi\)
\(234\) 0 0
\(235\) 7.00000 0.456630
\(236\) − 10.5616i − 0.687499i
\(237\) 7.43845 0.483179
\(238\) −18.2462 −1.18273
\(239\) − 13.3693i − 0.864789i −0.901684 0.432395i \(-0.857669\pi\)
0.901684 0.432395i \(-0.142331\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) − 19.8769i − 1.28038i −0.768215 0.640192i \(-0.778855\pi\)
0.768215 0.640192i \(-0.221145\pi\)
\(242\) 6.00000i 0.385695i
\(243\) 1.00000 0.0641500
\(244\) −6.00000 −0.384111
\(245\) 5.68466i 0.363180i
\(246\) 4.24621 0.270729
\(247\) 0 0
\(248\) 5.68466 0.360976
\(249\) − 1.12311i − 0.0711739i
\(250\) 1.00000 0.0632456
\(251\) −0.123106 −0.00777036 −0.00388518 0.999992i \(-0.501237\pi\)
−0.00388518 + 0.999992i \(0.501237\pi\)
\(252\) − 3.56155i − 0.224357i
\(253\) − 31.6847i − 1.99200i
\(254\) − 4.43845i − 0.278493i
\(255\) 5.12311i 0.320821i
\(256\) 1.00000 0.0625000
\(257\) 1.43845 0.0897279 0.0448639 0.998993i \(-0.485715\pi\)
0.0448639 + 0.998993i \(0.485715\pi\)
\(258\) 4.56155i 0.283990i
\(259\) −14.6847 −0.912460
\(260\) 0 0
\(261\) 6.56155 0.406150
\(262\) − 6.12311i − 0.378287i
\(263\) −13.0000 −0.801614 −0.400807 0.916162i \(-0.631270\pi\)
−0.400807 + 0.916162i \(0.631270\pi\)
\(264\) 4.12311 0.253760
\(265\) − 4.43845i − 0.272652i
\(266\) 12.6847i 0.777746i
\(267\) 1.80776i 0.110633i
\(268\) 14.2462i 0.870226i
\(269\) −3.36932 −0.205431 −0.102715 0.994711i \(-0.532753\pi\)
−0.102715 + 0.994711i \(0.532753\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 32.1771i 1.95462i 0.211818 + 0.977309i \(0.432062\pi\)
−0.211818 + 0.977309i \(0.567938\pi\)
\(272\) −5.12311 −0.310634
\(273\) 0 0
\(274\) −8.80776 −0.532096
\(275\) 4.12311i 0.248633i
\(276\) −7.68466 −0.462562
\(277\) 1.00000 0.0600842 0.0300421 0.999549i \(-0.490436\pi\)
0.0300421 + 0.999549i \(0.490436\pi\)
\(278\) − 8.43845i − 0.506104i
\(279\) − 5.68466i − 0.340332i
\(280\) 3.56155i 0.212843i
\(281\) − 0.246211i − 0.0146877i −0.999973 0.00734387i \(-0.997662\pi\)
0.999973 0.00734387i \(-0.00233765\pi\)
\(282\) 7.00000 0.416844
\(283\) −11.4384 −0.679945 −0.339973 0.940435i \(-0.610418\pi\)
−0.339973 + 0.940435i \(0.610418\pi\)
\(284\) − 4.87689i − 0.289390i
\(285\) 3.56155 0.210968
\(286\) 0 0
\(287\) −15.1231 −0.892689
\(288\) − 1.00000i − 0.0589256i
\(289\) 9.24621 0.543895
\(290\) −6.56155 −0.385308
\(291\) − 1.12311i − 0.0658376i
\(292\) 15.3693i 0.899421i
\(293\) − 24.9309i − 1.45648i −0.685324 0.728238i \(-0.740339\pi\)
0.685324 0.728238i \(-0.259661\pi\)
\(294\) 5.68466i 0.331536i
\(295\) 10.5616 0.614917
\(296\) −4.12311 −0.239651
\(297\) − 4.12311i − 0.239247i
\(298\) 22.8078 1.32122
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) − 16.2462i − 0.936416i
\(302\) 9.36932 0.539144
\(303\) 17.1231 0.983697
\(304\) 3.56155i 0.204269i
\(305\) − 6.00000i − 0.343559i
\(306\) 5.12311i 0.292868i
\(307\) − 15.6155i − 0.891225i −0.895226 0.445613i \(-0.852986\pi\)
0.895226 0.445613i \(-0.147014\pi\)
\(308\) −14.6847 −0.836736
\(309\) −0.438447 −0.0249424
\(310\) 5.68466i 0.322867i
\(311\) 18.7386 1.06257 0.531285 0.847193i \(-0.321709\pi\)
0.531285 + 0.847193i \(0.321709\pi\)
\(312\) 0 0
\(313\) 6.63068 0.374788 0.187394 0.982285i \(-0.439996\pi\)
0.187394 + 0.982285i \(0.439996\pi\)
\(314\) − 22.1231i − 1.24848i
\(315\) 3.56155 0.200671
\(316\) −7.43845 −0.418445
\(317\) 4.19224i 0.235459i 0.993046 + 0.117730i \(0.0375616\pi\)
−0.993046 + 0.117730i \(0.962438\pi\)
\(318\) − 4.43845i − 0.248896i
\(319\) − 27.0540i − 1.51473i
\(320\) 1.00000i 0.0559017i
\(321\) 2.00000 0.111629
\(322\) 27.3693 1.52523
\(323\) − 18.2462i − 1.01525i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −18.5616 −1.02803
\(327\) 20.2462i 1.11962i
\(328\) −4.24621 −0.234458
\(329\) −24.9309 −1.37448
\(330\) 4.12311i 0.226969i
\(331\) 18.7386i 1.02997i 0.857200 + 0.514984i \(0.172202\pi\)
−0.857200 + 0.514984i \(0.827798\pi\)
\(332\) 1.12311i 0.0616384i
\(333\) 4.12311i 0.225945i
\(334\) −20.3693 −1.11456
\(335\) −14.2462 −0.778354
\(336\) 3.56155i 0.194299i
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 0 0
\(339\) −3.68466 −0.200123
\(340\) − 5.12311i − 0.277839i
\(341\) −23.4384 −1.26926
\(342\) 3.56155 0.192587
\(343\) 4.68466i 0.252948i
\(344\) − 4.56155i − 0.245942i
\(345\) − 7.68466i − 0.413728i
\(346\) − 25.1771i − 1.35353i
\(347\) 9.12311 0.489754 0.244877 0.969554i \(-0.421252\pi\)
0.244877 + 0.969554i \(0.421252\pi\)
\(348\) −6.56155 −0.351736
\(349\) − 24.4924i − 1.31105i −0.755174 0.655525i \(-0.772448\pi\)
0.755174 0.655525i \(-0.227552\pi\)
\(350\) −3.56155 −0.190373
\(351\) 0 0
\(352\) −4.12311 −0.219762
\(353\) − 31.8617i − 1.69583i −0.530133 0.847915i \(-0.677858\pi\)
0.530133 0.847915i \(-0.322142\pi\)
\(354\) 10.5616 0.561340
\(355\) 4.87689 0.258839
\(356\) − 1.80776i − 0.0958113i
\(357\) − 18.2462i − 0.965692i
\(358\) − 0.315342i − 0.0166663i
\(359\) 4.87689i 0.257393i 0.991684 + 0.128696i \(0.0410792\pi\)
−0.991684 + 0.128696i \(0.958921\pi\)
\(360\) 1.00000 0.0527046
\(361\) 6.31534 0.332386
\(362\) 11.1231i 0.584617i
\(363\) −6.00000 −0.314918
\(364\) 0 0
\(365\) −15.3693 −0.804467
\(366\) − 6.00000i − 0.313625i
\(367\) −20.4924 −1.06970 −0.534848 0.844948i \(-0.679631\pi\)
−0.534848 + 0.844948i \(0.679631\pi\)
\(368\) 7.68466 0.400591
\(369\) 4.24621i 0.221049i
\(370\) − 4.12311i − 0.214350i
\(371\) 15.8078i 0.820698i
\(372\) 5.68466i 0.294736i
\(373\) −5.19224 −0.268844 −0.134422 0.990924i \(-0.542918\pi\)
−0.134422 + 0.990924i \(0.542918\pi\)
\(374\) 21.1231 1.09225
\(375\) 1.00000i 0.0516398i
\(376\) −7.00000 −0.360997
\(377\) 0 0
\(378\) 3.56155 0.183187
\(379\) 5.31534i 0.273031i 0.990638 + 0.136515i \(0.0435903\pi\)
−0.990638 + 0.136515i \(0.956410\pi\)
\(380\) −3.56155 −0.182704
\(381\) 4.43845 0.227389
\(382\) − 19.1231i − 0.978423i
\(383\) − 20.8078i − 1.06323i −0.846987 0.531614i \(-0.821586\pi\)
0.846987 0.531614i \(-0.178414\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) − 14.6847i − 0.748399i
\(386\) 0 0
\(387\) −4.56155 −0.231877
\(388\) 1.12311i 0.0570170i
\(389\) −19.0540 −0.966075 −0.483037 0.875600i \(-0.660467\pi\)
−0.483037 + 0.875600i \(0.660467\pi\)
\(390\) 0 0
\(391\) −39.3693 −1.99099
\(392\) − 5.68466i − 0.287119i
\(393\) 6.12311 0.308870
\(394\) 7.80776 0.393349
\(395\) − 7.43845i − 0.374269i
\(396\) 4.12311i 0.207194i
\(397\) − 11.8769i − 0.596084i −0.954553 0.298042i \(-0.903666\pi\)
0.954553 0.298042i \(-0.0963336\pi\)
\(398\) 11.1231i 0.557551i
\(399\) −12.6847 −0.635027
\(400\) −1.00000 −0.0500000
\(401\) 12.6847i 0.633442i 0.948519 + 0.316721i \(0.102582\pi\)
−0.948519 + 0.316721i \(0.897418\pi\)
\(402\) −14.2462 −0.710536
\(403\) 0 0
\(404\) −17.1231 −0.851906
\(405\) − 1.00000i − 0.0496904i
\(406\) 23.3693 1.15980
\(407\) 17.0000 0.842659
\(408\) − 5.12311i − 0.253632i
\(409\) − 1.80776i − 0.0893882i −0.999001 0.0446941i \(-0.985769\pi\)
0.999001 0.0446941i \(-0.0142313\pi\)
\(410\) − 4.24621i − 0.209705i
\(411\) − 8.80776i − 0.434455i
\(412\) 0.438447 0.0216007
\(413\) −37.6155 −1.85094
\(414\) − 7.68466i − 0.377680i
\(415\) −1.12311 −0.0551311
\(416\) 0 0
\(417\) 8.43845 0.413233
\(418\) − 14.6847i − 0.718250i
\(419\) −0.492423 −0.0240564 −0.0120282 0.999928i \(-0.503829\pi\)
−0.0120282 + 0.999928i \(0.503829\pi\)
\(420\) −3.56155 −0.173786
\(421\) − 0.492423i − 0.0239992i −0.999928 0.0119996i \(-0.996180\pi\)
0.999928 0.0119996i \(-0.00381968\pi\)
\(422\) − 6.93087i − 0.337389i
\(423\) 7.00000i 0.340352i
\(424\) 4.43845i 0.215550i
\(425\) 5.12311 0.248507
\(426\) 4.87689 0.236286
\(427\) 21.3693i 1.03413i
\(428\) −2.00000 −0.0966736
\(429\) 0 0
\(430\) 4.56155 0.219978
\(431\) − 26.7386i − 1.28795i −0.765045 0.643977i \(-0.777283\pi\)
0.765045 0.643977i \(-0.222717\pi\)
\(432\) 1.00000 0.0481125
\(433\) −30.7386 −1.47720 −0.738602 0.674141i \(-0.764514\pi\)
−0.738602 + 0.674141i \(0.764514\pi\)
\(434\) − 20.2462i − 0.971849i
\(435\) − 6.56155i − 0.314602i
\(436\) − 20.2462i − 0.969618i
\(437\) 27.3693i 1.30925i
\(438\) −15.3693 −0.734374
\(439\) −16.8769 −0.805490 −0.402745 0.915312i \(-0.631944\pi\)
−0.402745 + 0.915312i \(0.631944\pi\)
\(440\) − 4.12311i − 0.196561i
\(441\) −5.68466 −0.270698
\(442\) 0 0
\(443\) −4.87689 −0.231708 −0.115854 0.993266i \(-0.536961\pi\)
−0.115854 + 0.993266i \(0.536961\pi\)
\(444\) − 4.12311i − 0.195674i
\(445\) 1.80776 0.0856962
\(446\) 26.3002 1.24535
\(447\) 22.8078i 1.07877i
\(448\) − 3.56155i − 0.168268i
\(449\) 25.1771i 1.18818i 0.804399 + 0.594090i \(0.202488\pi\)
−0.804399 + 0.594090i \(0.797512\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 17.5076 0.824400
\(452\) 3.68466 0.173312
\(453\) 9.36932i 0.440209i
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) −3.56155 −0.166785
\(457\) − 3.75379i − 0.175595i −0.996138 0.0877974i \(-0.972017\pi\)
0.996138 0.0877974i \(-0.0279828\pi\)
\(458\) −7.75379 −0.362311
\(459\) −5.12311 −0.239126
\(460\) 7.68466i 0.358299i
\(461\) − 7.05398i − 0.328536i −0.986416 0.164268i \(-0.947474\pi\)
0.986416 0.164268i \(-0.0525262\pi\)
\(462\) − 14.6847i − 0.683192i
\(463\) 33.6155i 1.56225i 0.624377 + 0.781123i \(0.285353\pi\)
−0.624377 + 0.781123i \(0.714647\pi\)
\(464\) 6.56155 0.304612
\(465\) −5.68466 −0.263620
\(466\) − 17.6847i − 0.819226i
\(467\) −39.8617 −1.84458 −0.922291 0.386497i \(-0.873685\pi\)
−0.922291 + 0.386497i \(0.873685\pi\)
\(468\) 0 0
\(469\) 50.7386 2.34289
\(470\) − 7.00000i − 0.322886i
\(471\) 22.1231 1.01938
\(472\) −10.5616 −0.486135
\(473\) 18.8078i 0.864782i
\(474\) − 7.43845i − 0.341659i
\(475\) − 3.56155i − 0.163415i
\(476\) 18.2462i 0.836314i
\(477\) 4.43845 0.203223
\(478\) −13.3693 −0.611498
\(479\) − 21.7538i − 0.993956i −0.867763 0.496978i \(-0.834443\pi\)
0.867763 0.496978i \(-0.165557\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −19.8769 −0.905368
\(483\) 27.3693i 1.24535i
\(484\) 6.00000 0.272727
\(485\) −1.12311 −0.0509976
\(486\) − 1.00000i − 0.0453609i
\(487\) 24.0540i 1.08999i 0.838439 + 0.544995i \(0.183468\pi\)
−0.838439 + 0.544995i \(0.816532\pi\)
\(488\) 6.00000i 0.271607i
\(489\) − 18.5616i − 0.839382i
\(490\) 5.68466 0.256807
\(491\) 21.5616 0.973059 0.486530 0.873664i \(-0.338263\pi\)
0.486530 + 0.873664i \(0.338263\pi\)
\(492\) − 4.24621i − 0.191434i
\(493\) −33.6155 −1.51397
\(494\) 0 0
\(495\) −4.12311 −0.185320
\(496\) − 5.68466i − 0.255249i
\(497\) −17.3693 −0.779120
\(498\) −1.12311 −0.0503276
\(499\) − 4.00000i − 0.179065i −0.995984 0.0895323i \(-0.971463\pi\)
0.995984 0.0895323i \(-0.0285372\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) − 20.3693i − 0.910034i
\(502\) 0.123106i 0.00549447i
\(503\) −5.94602 −0.265120 −0.132560 0.991175i \(-0.542320\pi\)
−0.132560 + 0.991175i \(0.542320\pi\)
\(504\) −3.56155 −0.158644
\(505\) − 17.1231i − 0.761968i
\(506\) −31.6847 −1.40855
\(507\) 0 0
\(508\) −4.43845 −0.196924
\(509\) 29.5464i 1.30962i 0.755793 + 0.654811i \(0.227252\pi\)
−0.755793 + 0.654811i \(0.772748\pi\)
\(510\) 5.12311 0.226855
\(511\) 54.7386 2.42149
\(512\) − 1.00000i − 0.0441942i
\(513\) 3.56155i 0.157246i
\(514\) − 1.43845i − 0.0634472i
\(515\) 0.438447i 0.0193203i
\(516\) 4.56155 0.200811
\(517\) 28.8617 1.26934
\(518\) 14.6847i 0.645207i
\(519\) 25.1771 1.10515
\(520\) 0 0
\(521\) −18.6847 −0.818590 −0.409295 0.912402i \(-0.634225\pi\)
−0.409295 + 0.912402i \(0.634225\pi\)
\(522\) − 6.56155i − 0.287191i
\(523\) −9.19224 −0.401948 −0.200974 0.979597i \(-0.564411\pi\)
−0.200974 + 0.979597i \(0.564411\pi\)
\(524\) −6.12311 −0.267489
\(525\) − 3.56155i − 0.155439i
\(526\) 13.0000i 0.566827i
\(527\) 29.1231i 1.26862i
\(528\) − 4.12311i − 0.179435i
\(529\) 36.0540 1.56756
\(530\) −4.43845 −0.192794
\(531\) 10.5616i 0.458332i
\(532\) 12.6847 0.549950
\(533\) 0 0
\(534\) 1.80776 0.0782296
\(535\) − 2.00000i − 0.0864675i
\(536\) 14.2462 0.615343
\(537\) 0.315342 0.0136080
\(538\) 3.36932i 0.145262i
\(539\) 23.4384i 1.00957i
\(540\) 1.00000i 0.0430331i
\(541\) − 2.63068i − 0.113102i −0.998400 0.0565510i \(-0.981990\pi\)
0.998400 0.0565510i \(-0.0180103\pi\)
\(542\) 32.1771 1.38212
\(543\) −11.1231 −0.477338
\(544\) 5.12311i 0.219651i
\(545\) 20.2462 0.867252
\(546\) 0 0
\(547\) 35.6155 1.52281 0.761405 0.648276i \(-0.224510\pi\)
0.761405 + 0.648276i \(0.224510\pi\)
\(548\) 8.80776i 0.376249i
\(549\) 6.00000 0.256074
\(550\) 4.12311 0.175810
\(551\) 23.3693i 0.995566i
\(552\) 7.68466i 0.327081i
\(553\) 26.4924i 1.12657i
\(554\) − 1.00000i − 0.0424859i
\(555\) 4.12311 0.175016
\(556\) −8.43845 −0.357870
\(557\) − 4.43845i − 0.188063i −0.995569 0.0940315i \(-0.970025\pi\)
0.995569 0.0940315i \(-0.0299754\pi\)
\(558\) −5.68466 −0.240651
\(559\) 0 0
\(560\) 3.56155 0.150503
\(561\) 21.1231i 0.891818i
\(562\) −0.246211 −0.0103858
\(563\) −32.9848 −1.39015 −0.695073 0.718939i \(-0.744628\pi\)
−0.695073 + 0.718939i \(0.744628\pi\)
\(564\) − 7.00000i − 0.294753i
\(565\) 3.68466i 0.155015i
\(566\) 11.4384i 0.480794i
\(567\) 3.56155i 0.149571i
\(568\) −4.87689 −0.204630
\(569\) 19.1771 0.803945 0.401973 0.915652i \(-0.368325\pi\)
0.401973 + 0.915652i \(0.368325\pi\)
\(570\) − 3.56155i − 0.149177i
\(571\) −11.3153 −0.473532 −0.236766 0.971567i \(-0.576088\pi\)
−0.236766 + 0.971567i \(0.576088\pi\)
\(572\) 0 0
\(573\) 19.1231 0.798879
\(574\) 15.1231i 0.631226i
\(575\) −7.68466 −0.320472
\(576\) −1.00000 −0.0416667
\(577\) − 8.73863i − 0.363794i −0.983318 0.181897i \(-0.941776\pi\)
0.983318 0.181897i \(-0.0582238\pi\)
\(578\) − 9.24621i − 0.384592i
\(579\) 0 0
\(580\) 6.56155i 0.272454i
\(581\) 4.00000 0.165948
\(582\) −1.12311 −0.0465542
\(583\) − 18.3002i − 0.757916i
\(584\) 15.3693 0.635987
\(585\) 0 0
\(586\) −24.9309 −1.02988
\(587\) 23.7538i 0.980424i 0.871603 + 0.490212i \(0.163081\pi\)
−0.871603 + 0.490212i \(0.836919\pi\)
\(588\) 5.68466 0.234431
\(589\) 20.2462 0.834231
\(590\) − 10.5616i − 0.434812i
\(591\) 7.80776i 0.321168i
\(592\) 4.12311i 0.169459i
\(593\) − 12.1771i − 0.500053i −0.968239 0.250026i \(-0.919561\pi\)
0.968239 0.250026i \(-0.0804393\pi\)
\(594\) −4.12311 −0.169173
\(595\) −18.2462 −0.748022
\(596\) − 22.8078i − 0.934242i
\(597\) −11.1231 −0.455238
\(598\) 0 0
\(599\) −14.0000 −0.572024 −0.286012 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) −35.9848 −1.46785 −0.733926 0.679229i \(-0.762314\pi\)
−0.733926 + 0.679229i \(0.762314\pi\)
\(602\) −16.2462 −0.662146
\(603\) − 14.2462i − 0.580151i
\(604\) − 9.36932i − 0.381232i
\(605\) 6.00000i 0.243935i
\(606\) − 17.1231i − 0.695579i
\(607\) −29.4233 −1.19425 −0.597127 0.802146i \(-0.703691\pi\)
−0.597127 + 0.802146i \(0.703691\pi\)
\(608\) 3.56155 0.144440
\(609\) 23.3693i 0.946973i
\(610\) −6.00000 −0.242933
\(611\) 0 0
\(612\) 5.12311 0.207089
\(613\) − 28.1231i − 1.13588i −0.823069 0.567941i \(-0.807740\pi\)
0.823069 0.567941i \(-0.192260\pi\)
\(614\) −15.6155 −0.630191
\(615\) 4.24621 0.171224
\(616\) 14.6847i 0.591662i
\(617\) 41.6847i 1.67816i 0.544007 + 0.839081i \(0.316906\pi\)
−0.544007 + 0.839081i \(0.683094\pi\)
\(618\) 0.438447i 0.0176369i
\(619\) − 16.4384i − 0.660717i −0.943856 0.330358i \(-0.892830\pi\)
0.943856 0.330358i \(-0.107170\pi\)
\(620\) 5.68466 0.228301
\(621\) 7.68466 0.308375
\(622\) − 18.7386i − 0.751351i
\(623\) −6.43845 −0.257951
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 6.63068i − 0.265015i
\(627\) 14.6847 0.586449
\(628\) −22.1231 −0.882808
\(629\) − 21.1231i − 0.842233i
\(630\) − 3.56155i − 0.141896i
\(631\) − 30.2462i − 1.20408i −0.798465 0.602041i \(-0.794354\pi\)
0.798465 0.602041i \(-0.205646\pi\)
\(632\) 7.43845i 0.295886i
\(633\) 6.93087 0.275477
\(634\) 4.19224 0.166495
\(635\) − 4.43845i − 0.176134i
\(636\) −4.43845 −0.175996
\(637\) 0 0
\(638\) −27.0540 −1.07108
\(639\) 4.87689i 0.192927i
\(640\) 1.00000 0.0395285
\(641\) 15.5616 0.614644 0.307322 0.951606i \(-0.400567\pi\)
0.307322 + 0.951606i \(0.400567\pi\)
\(642\) − 2.00000i − 0.0789337i
\(643\) − 38.2462i − 1.50828i −0.656712 0.754142i \(-0.728053\pi\)
0.656712 0.754142i \(-0.271947\pi\)
\(644\) − 27.3693i − 1.07850i
\(645\) 4.56155i 0.179611i
\(646\) −18.2462 −0.717888
\(647\) 2.05398 0.0807501 0.0403751 0.999185i \(-0.487145\pi\)
0.0403751 + 0.999185i \(0.487145\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 43.5464 1.70935
\(650\) 0 0
\(651\) 20.2462 0.793512
\(652\) 18.5616i 0.726927i
\(653\) −40.4384 −1.58248 −0.791239 0.611507i \(-0.790564\pi\)
−0.791239 + 0.611507i \(0.790564\pi\)
\(654\) 20.2462 0.791690
\(655\) − 6.12311i − 0.239250i
\(656\) 4.24621i 0.165787i
\(657\) − 15.3693i − 0.599614i
\(658\) 24.9309i 0.971906i
\(659\) 31.0540 1.20969 0.604846 0.796343i \(-0.293235\pi\)
0.604846 + 0.796343i \(0.293235\pi\)
\(660\) 4.12311 0.160492
\(661\) − 11.6155i − 0.451792i −0.974151 0.225896i \(-0.927469\pi\)
0.974151 0.225896i \(-0.0725309\pi\)
\(662\) 18.7386 0.728298
\(663\) 0 0
\(664\) 1.12311 0.0435850
\(665\) 12.6847i 0.491890i
\(666\) 4.12311 0.159767
\(667\) 50.4233 1.95240
\(668\) 20.3693i 0.788113i
\(669\) 26.3002i 1.01682i
\(670\) 14.2462i 0.550379i
\(671\) − 24.7386i − 0.955024i
\(672\) 3.56155 0.137390
\(673\) −42.2462 −1.62847 −0.814236 0.580534i \(-0.802844\pi\)
−0.814236 + 0.580534i \(0.802844\pi\)
\(674\) − 6.00000i − 0.231111i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −28.8769 −1.10983 −0.554915 0.831907i \(-0.687249\pi\)
−0.554915 + 0.831907i \(0.687249\pi\)
\(678\) 3.68466i 0.141508i
\(679\) 4.00000 0.153506
\(680\) −5.12311 −0.196462
\(681\) 20.0000i 0.766402i
\(682\) 23.4384i 0.897505i
\(683\) 8.87689i 0.339665i 0.985473 + 0.169832i \(0.0543227\pi\)
−0.985473 + 0.169832i \(0.945677\pi\)
\(684\) − 3.56155i − 0.136179i
\(685\) −8.80776 −0.336527
\(686\) 4.68466 0.178861
\(687\) − 7.75379i − 0.295825i
\(688\) −4.56155 −0.173908
\(689\) 0 0
\(690\) −7.68466 −0.292550
\(691\) 16.4384i 0.625348i 0.949860 + 0.312674i \(0.101225\pi\)
−0.949860 + 0.312674i \(0.898775\pi\)
\(692\) −25.1771 −0.957089
\(693\) 14.6847 0.557824
\(694\) − 9.12311i − 0.346308i
\(695\) − 8.43845i − 0.320089i
\(696\) 6.56155i 0.248715i
\(697\) − 21.7538i − 0.823984i
\(698\) −24.4924 −0.927052
\(699\) 17.6847 0.668895
\(700\) 3.56155i 0.134614i
\(701\) −17.3002 −0.653419 −0.326710 0.945125i \(-0.605940\pi\)
−0.326710 + 0.945125i \(0.605940\pi\)
\(702\) 0 0
\(703\) −14.6847 −0.553842
\(704\) 4.12311i 0.155395i
\(705\) 7.00000 0.263635
\(706\) −31.8617 −1.19913
\(707\) 60.9848i 2.29357i
\(708\) − 10.5616i − 0.396927i
\(709\) 17.7538i 0.666758i 0.942793 + 0.333379i \(0.108189\pi\)
−0.942793 + 0.333379i \(0.891811\pi\)
\(710\) − 4.87689i − 0.183027i
\(711\) 7.43845 0.278964
\(712\) −1.80776 −0.0677488
\(713\) − 43.6847i − 1.63600i
\(714\) −18.2462 −0.682847
\(715\) 0 0
\(716\) −0.315342 −0.0117849
\(717\) − 13.3693i − 0.499286i
\(718\) 4.87689 0.182004
\(719\) 20.9848 0.782603 0.391301 0.920263i \(-0.372025\pi\)
0.391301 + 0.920263i \(0.372025\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) − 1.56155i − 0.0581553i
\(722\) − 6.31534i − 0.235033i
\(723\) − 19.8769i − 0.739230i
\(724\) 11.1231 0.413387
\(725\) −6.56155 −0.243690
\(726\) 6.00000i 0.222681i
\(727\) 23.4233 0.868722 0.434361 0.900739i \(-0.356974\pi\)
0.434361 + 0.900739i \(0.356974\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 15.3693i 0.568844i
\(731\) 23.3693 0.864345
\(732\) −6.00000 −0.221766
\(733\) − 48.9309i − 1.80730i −0.428269 0.903651i \(-0.640876\pi\)
0.428269 0.903651i \(-0.359124\pi\)
\(734\) 20.4924i 0.756389i
\(735\) 5.68466i 0.209682i
\(736\) − 7.68466i − 0.283260i
\(737\) −58.7386 −2.16367
\(738\) 4.24621 0.156305
\(739\) 42.5464i 1.56509i 0.622591 + 0.782547i \(0.286080\pi\)
−0.622591 + 0.782547i \(0.713920\pi\)
\(740\) −4.12311 −0.151568
\(741\) 0 0
\(742\) 15.8078 0.580321
\(743\) 32.1771i 1.18046i 0.807234 + 0.590231i \(0.200963\pi\)
−0.807234 + 0.590231i \(0.799037\pi\)
\(744\) 5.68466 0.208410
\(745\) 22.8078 0.835612
\(746\) 5.19224i 0.190101i
\(747\) − 1.12311i − 0.0410923i
\(748\) − 21.1231i − 0.772337i
\(749\) 7.12311i 0.260273i
\(750\) 1.00000 0.0365148
\(751\) 3.19224 0.116486 0.0582432 0.998302i \(-0.481450\pi\)
0.0582432 + 0.998302i \(0.481450\pi\)
\(752\) 7.00000i 0.255264i
\(753\) −0.123106 −0.00448622
\(754\) 0 0
\(755\) 9.36932 0.340984
\(756\) − 3.56155i − 0.129532i
\(757\) 33.4233 1.21479 0.607395 0.794400i \(-0.292215\pi\)
0.607395 + 0.794400i \(0.292215\pi\)
\(758\) 5.31534 0.193062
\(759\) − 31.6847i − 1.15008i
\(760\) 3.56155i 0.129191i
\(761\) 10.9309i 0.396244i 0.980177 + 0.198122i \(0.0634842\pi\)
−0.980177 + 0.198122i \(0.936516\pi\)
\(762\) − 4.43845i − 0.160788i
\(763\) −72.1080 −2.61048
\(764\) −19.1231 −0.691850
\(765\) 5.12311i 0.185226i
\(766\) −20.8078 −0.751815
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 45.6847i 1.64743i 0.567003 + 0.823715i \(0.308103\pi\)
−0.567003 + 0.823715i \(0.691897\pi\)
\(770\) −14.6847 −0.529198
\(771\) 1.43845 0.0518044
\(772\) 0 0
\(773\) 21.1771i 0.761687i 0.924640 + 0.380843i \(0.124366\pi\)
−0.924640 + 0.380843i \(0.875634\pi\)
\(774\) 4.56155i 0.163962i
\(775\) 5.68466i 0.204199i
\(776\) 1.12311 0.0403171
\(777\) −14.6847 −0.526809
\(778\) 19.0540i 0.683118i
\(779\) −15.1231 −0.541841
\(780\) 0 0
\(781\) 20.1080 0.719519
\(782\) 39.3693i 1.40784i
\(783\) 6.56155 0.234491
\(784\) −5.68466 −0.203024
\(785\) − 22.1231i − 0.789607i
\(786\) − 6.12311i − 0.218404i
\(787\) 43.6847i 1.55719i 0.627527 + 0.778595i \(0.284067\pi\)
−0.627527 + 0.778595i \(0.715933\pi\)
\(788\) − 7.80776i − 0.278140i
\(789\) −13.0000 −0.462812
\(790\) −7.43845 −0.264648
\(791\) − 13.1231i − 0.466604i
\(792\) 4.12311 0.146508
\(793\) 0 0
\(794\) −11.8769 −0.421495
\(795\) − 4.43845i − 0.157415i
\(796\) 11.1231 0.394248
\(797\) −4.87689 −0.172748 −0.0863742 0.996263i \(-0.527528\pi\)
−0.0863742 + 0.996263i \(0.527528\pi\)
\(798\) 12.6847i 0.449032i
\(799\) − 35.8617i − 1.26870i
\(800\) 1.00000i 0.0353553i
\(801\) 1.80776i 0.0638742i
\(802\) 12.6847 0.447911
\(803\) −63.3693 −2.23625
\(804\) 14.2462i 0.502425i
\(805\) 27.3693 0.964642
\(806\) 0 0
\(807\) −3.36932 −0.118606
\(808\) 17.1231i 0.602389i
\(809\) 18.4924 0.650159 0.325079 0.945687i \(-0.394609\pi\)
0.325079 + 0.945687i \(0.394609\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 22.5464i 0.791711i 0.918313 + 0.395856i \(0.129552\pi\)
−0.918313 + 0.395856i \(0.870448\pi\)
\(812\) − 23.3693i − 0.820102i
\(813\) 32.1771i 1.12850i
\(814\) − 17.0000i − 0.595850i
\(815\) −18.5616 −0.650183
\(816\) −5.12311 −0.179345
\(817\) − 16.2462i − 0.568383i
\(818\) −1.80776 −0.0632070
\(819\) 0 0
\(820\) −4.24621 −0.148284
\(821\) 12.5616i 0.438401i 0.975680 + 0.219201i \(0.0703449\pi\)
−0.975680 + 0.219201i \(0.929655\pi\)
\(822\) −8.80776 −0.307206
\(823\) 32.0540 1.11733 0.558666 0.829393i \(-0.311313\pi\)
0.558666 + 0.829393i \(0.311313\pi\)
\(824\) − 0.438447i − 0.0152740i
\(825\) 4.12311i 0.143548i
\(826\) 37.6155i 1.30881i
\(827\) − 11.7538i − 0.408719i −0.978896 0.204360i \(-0.934489\pi\)
0.978896 0.204360i \(-0.0655112\pi\)
\(828\) −7.68466 −0.267060
\(829\) −53.6155 −1.86214 −0.931072 0.364835i \(-0.881125\pi\)
−0.931072 + 0.364835i \(0.881125\pi\)
\(830\) 1.12311i 0.0389836i
\(831\) 1.00000 0.0346896
\(832\) 0 0
\(833\) 29.1231 1.00906
\(834\) − 8.43845i − 0.292200i
\(835\) −20.3693 −0.704909
\(836\) −14.6847 −0.507880
\(837\) − 5.68466i − 0.196491i
\(838\) 0.492423i 0.0170105i
\(839\) − 12.7386i − 0.439786i −0.975524 0.219893i \(-0.929429\pi\)
0.975524 0.219893i \(-0.0705709\pi\)
\(840\) 3.56155i 0.122885i
\(841\) 14.0540 0.484620
\(842\) −0.492423 −0.0169700
\(843\) − 0.246211i − 0.00847997i
\(844\) −6.93087 −0.238570
\(845\) 0 0
\(846\) 7.00000 0.240665
\(847\) − 21.3693i − 0.734258i
\(848\) 4.43845 0.152417
\(849\) −11.4384 −0.392566
\(850\) − 5.12311i − 0.175721i
\(851\) 31.6847i 1.08614i
\(852\) − 4.87689i − 0.167080i
\(853\) − 51.3002i − 1.75648i −0.478216 0.878242i \(-0.658716\pi\)
0.478216 0.878242i \(-0.341284\pi\)
\(854\) 21.3693 0.731243
\(855\) 3.56155 0.121803
\(856\) 2.00000i 0.0683586i
\(857\) 26.8078 0.915736 0.457868 0.889020i \(-0.348613\pi\)
0.457868 + 0.889020i \(0.348613\pi\)
\(858\) 0 0
\(859\) −46.7926 −1.59654 −0.798272 0.602298i \(-0.794252\pi\)
−0.798272 + 0.602298i \(0.794252\pi\)
\(860\) − 4.56155i − 0.155548i
\(861\) −15.1231 −0.515394
\(862\) −26.7386 −0.910721
\(863\) − 6.56155i − 0.223358i −0.993744 0.111679i \(-0.964377\pi\)
0.993744 0.111679i \(-0.0356228\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) − 25.1771i − 0.856046i
\(866\) 30.7386i 1.04454i
\(867\) 9.24621 0.314018
\(868\) −20.2462 −0.687201
\(869\) − 30.6695i − 1.04039i
\(870\) −6.56155 −0.222457
\(871\) 0 0
\(872\) −20.2462 −0.685623
\(873\) − 1.12311i − 0.0380114i
\(874\) 27.3693 0.925781
\(875\) −3.56155 −0.120402
\(876\) 15.3693i 0.519281i
\(877\) − 12.5616i − 0.424173i −0.977251 0.212087i \(-0.931974\pi\)
0.977251 0.212087i \(-0.0680259\pi\)
\(878\) 16.8769i 0.569568i
\(879\) − 24.9309i − 0.840897i
\(880\) −4.12311 −0.138990
\(881\) 20.4384 0.688589 0.344294 0.938862i \(-0.388118\pi\)
0.344294 + 0.938862i \(0.388118\pi\)
\(882\) 5.68466i 0.191412i
\(883\) −26.1771 −0.880929 −0.440464 0.897770i \(-0.645186\pi\)
−0.440464 + 0.897770i \(0.645186\pi\)
\(884\) 0 0
\(885\) 10.5616 0.355023
\(886\) 4.87689i 0.163842i
\(887\) 41.1080 1.38027 0.690135 0.723681i \(-0.257551\pi\)
0.690135 + 0.723681i \(0.257551\pi\)
\(888\) −4.12311 −0.138362
\(889\) 15.8078i 0.530175i
\(890\) − 1.80776i − 0.0605964i
\(891\) − 4.12311i − 0.138129i
\(892\) − 26.3002i − 0.880595i
\(893\) −24.9309 −0.834280
\(894\) 22.8078 0.762806
\(895\) − 0.315342i − 0.0105407i
\(896\) −3.56155 −0.118983
\(897\) 0 0
\(898\) 25.1771 0.840170
\(899\) − 37.3002i − 1.24403i
\(900\) 1.00000 0.0333333
\(901\) −22.7386 −0.757534
\(902\) − 17.5076i − 0.582939i
\(903\) − 16.2462i − 0.540640i
\(904\) − 3.68466i − 0.122550i
\(905\) 11.1231i 0.369745i
\(906\) 9.36932 0.311275
\(907\) −40.9157 −1.35858 −0.679292 0.733868i \(-0.737713\pi\)
−0.679292 + 0.733868i \(0.737713\pi\)
\(908\) − 20.0000i − 0.663723i
\(909\) 17.1231 0.567938
\(910\) 0 0
\(911\) −43.3693 −1.43689 −0.718445 0.695584i \(-0.755146\pi\)
−0.718445 + 0.695584i \(0.755146\pi\)
\(912\) 3.56155i 0.117935i
\(913\) −4.63068 −0.153253
\(914\) −3.75379 −0.124164
\(915\) − 6.00000i − 0.198354i
\(916\) 7.75379i 0.256192i
\(917\) 21.8078i 0.720156i
\(918\) 5.12311i 0.169088i
\(919\) 41.3693 1.36465 0.682324 0.731050i \(-0.260969\pi\)
0.682324 + 0.731050i \(0.260969\pi\)
\(920\) 7.68466 0.253356
\(921\) − 15.6155i − 0.514549i
\(922\) −7.05398 −0.232310
\(923\) 0 0
\(924\) −14.6847 −0.483090
\(925\) − 4.12311i − 0.135567i
\(926\) 33.6155 1.10467
\(927\) −0.438447 −0.0144005
\(928\) − 6.56155i − 0.215394i
\(929\) 16.8769i 0.553713i 0.960911 + 0.276856i \(0.0892927\pi\)
−0.960911 + 0.276856i \(0.910707\pi\)
\(930\) 5.68466i 0.186407i
\(931\) − 20.2462i − 0.663543i
\(932\) −17.6847 −0.579280
\(933\) 18.7386 0.613475
\(934\) 39.8617i 1.30432i
\(935\) 21.1231 0.690799
\(936\) 0 0
\(937\) −29.2311 −0.954937 −0.477468 0.878649i \(-0.658446\pi\)
−0.477468 + 0.878649i \(0.658446\pi\)
\(938\) − 50.7386i − 1.65668i
\(939\) 6.63068 0.216384
\(940\) −7.00000 −0.228315
\(941\) 0.630683i 0.0205597i 0.999947 + 0.0102798i \(0.00327223\pi\)
−0.999947 + 0.0102798i \(0.996728\pi\)
\(942\) − 22.1231i − 0.720810i
\(943\) 32.6307i 1.06260i
\(944\) 10.5616i 0.343749i
\(945\) 3.56155 0.115857
\(946\) 18.8078 0.611493
\(947\) 13.8617i 0.450446i 0.974307 + 0.225223i \(0.0723111\pi\)
−0.974307 + 0.225223i \(0.927689\pi\)
\(948\) −7.43845 −0.241590
\(949\) 0 0
\(950\) −3.56155 −0.115552
\(951\) 4.19224i 0.135943i
\(952\) 18.2462 0.591363
\(953\) −5.82292 −0.188623 −0.0943114 0.995543i \(-0.530065\pi\)
−0.0943114 + 0.995543i \(0.530065\pi\)
\(954\) − 4.43845i − 0.143700i
\(955\) − 19.1231i − 0.618809i
\(956\) 13.3693i 0.432395i
\(957\) − 27.0540i − 0.874531i
\(958\) −21.7538 −0.702833
\(959\) 31.3693 1.01297
\(960\) 1.00000i 0.0322749i
\(961\) −1.31534 −0.0424304
\(962\) 0 0
\(963\) 2.00000 0.0644491
\(964\) 19.8769i 0.640192i
\(965\) 0 0
\(966\) 27.3693 0.880593
\(967\) − 3.31534i − 0.106614i −0.998578 0.0533071i \(-0.983024\pi\)
0.998578 0.0533071i \(-0.0169762\pi\)
\(968\) − 6.00000i − 0.192847i
\(969\) − 18.2462i − 0.586153i
\(970\) 1.12311i 0.0360607i
\(971\) −16.6847 −0.535436 −0.267718 0.963497i \(-0.586270\pi\)
−0.267718 + 0.963497i \(0.586270\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 30.0540i 0.963486i
\(974\) 24.0540 0.770739
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) − 5.30019i − 0.169568i −0.996399 0.0847840i \(-0.972980\pi\)
0.996399 0.0847840i \(-0.0270200\pi\)
\(978\) −18.5616 −0.593533
\(979\) 7.45360 0.238218
\(980\) − 5.68466i − 0.181590i
\(981\) 20.2462i 0.646412i
\(982\) − 21.5616i − 0.688057i
\(983\) 23.8769i 0.761555i 0.924667 + 0.380777i \(0.124344\pi\)
−0.924667 + 0.380777i \(0.875656\pi\)
\(984\) −4.24621 −0.135364
\(985\) 7.80776 0.248776
\(986\) 33.6155i 1.07054i
\(987\) −24.9309 −0.793558
\(988\) 0 0
\(989\) −35.0540 −1.11465
\(990\) 4.12311i 0.131041i
\(991\) −12.4233 −0.394639 −0.197319 0.980339i \(-0.563224\pi\)
−0.197319 + 0.980339i \(0.563224\pi\)
\(992\) −5.68466 −0.180488
\(993\) 18.7386i 0.594653i
\(994\) 17.3693i 0.550921i
\(995\) 11.1231i 0.352626i
\(996\) 1.12311i 0.0355870i
\(997\) 32.5464 1.03075 0.515377 0.856963i \(-0.327652\pi\)
0.515377 + 0.856963i \(0.327652\pi\)
\(998\) −4.00000 −0.126618
\(999\) 4.12311i 0.130449i
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.r.1351.2 4
13.5 odd 4 5070.2.a.bb.1.1 2
13.7 odd 12 390.2.i.g.211.1 yes 4
13.8 odd 4 5070.2.a.bi.1.2 2
13.11 odd 12 390.2.i.g.61.1 4
13.12 even 2 inner 5070.2.b.r.1351.3 4
39.11 even 12 1170.2.i.o.451.1 4
39.20 even 12 1170.2.i.o.991.1 4
65.7 even 12 1950.2.z.n.1849.3 8
65.24 odd 12 1950.2.i.bi.451.2 4
65.33 even 12 1950.2.z.n.1849.2 8
65.37 even 12 1950.2.z.n.1699.2 8
65.59 odd 12 1950.2.i.bi.601.2 4
65.63 even 12 1950.2.z.n.1699.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.g.61.1 4 13.11 odd 12
390.2.i.g.211.1 yes 4 13.7 odd 12
1170.2.i.o.451.1 4 39.11 even 12
1170.2.i.o.991.1 4 39.20 even 12
1950.2.i.bi.451.2 4 65.24 odd 12
1950.2.i.bi.601.2 4 65.59 odd 12
1950.2.z.n.1699.2 8 65.37 even 12
1950.2.z.n.1699.3 8 65.63 even 12
1950.2.z.n.1849.2 8 65.33 even 12
1950.2.z.n.1849.3 8 65.7 even 12
5070.2.a.bb.1.1 2 13.5 odd 4
5070.2.a.bi.1.2 2 13.8 odd 4
5070.2.b.r.1351.2 4 1.1 even 1 trivial
5070.2.b.r.1351.3 4 13.12 even 2 inner