Properties

Label 1280.3.e.f.639.6
Level $1280$
Weight $3$
Character 1280.639
Analytic conductor $34.877$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1280.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.8774738381\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
Defining polynomial: \(x^{6} + 9 x^{4} + 14 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 639.6
Root \(2.65109i\) of defining polynomial
Character \(\chi\) \(=\) 1280.639
Dual form 1280.3.e.f.639.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.30219i q^{3} +(-4.75441 + 1.54778i) q^{5} +0.206625 q^{7} -19.1132 q^{9} +O(q^{10})\) \(q+5.30219i q^{3} +(-4.75441 + 1.54778i) q^{5} +0.206625 q^{7} -19.1132 q^{9} -15.0176 q^{11} -11.6999 q^{13} +(-8.20662 - 25.2087i) q^{15} +18.1911i q^{17} -19.3999 q^{19} +1.09556i q^{21} +27.2242 q^{23} +(20.2087 - 14.7176i) q^{25} -53.6220i q^{27} +44.4175i q^{29} -20.3822i q^{31} -79.6262i q^{33} +(-0.982377 + 0.319810i) q^{35} +18.1089 q^{37} -62.0352i q^{39} +32.3043 q^{41} +4.06244i q^{43} +(90.8718 - 29.5830i) q^{45} +5.37588 q^{47} -48.9573 q^{49} -96.4527 q^{51} +79.1703 q^{53} +(71.3999 - 23.2440i) q^{55} -102.862i q^{57} -83.3999 q^{59} -36.7486i q^{61} -3.94925 q^{63} +(55.6262 - 18.1089i) q^{65} -4.51518i q^{67} +144.348i q^{69} +41.6530i q^{71} +41.5910i q^{73} +(78.0352 + 107.151i) q^{75} -3.10301 q^{77} +15.5473i q^{79} +112.295 q^{81} -50.9862i q^{83} +(-28.1559 - 86.4880i) q^{85} -235.510 q^{87} -10.8885 q^{89} -2.41749 q^{91} +108.070 q^{93} +(92.2349 - 30.0268i) q^{95} +12.1559i q^{97} +287.035 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{5} - 12 q^{7} - 18 q^{9} + O(q^{10}) \) \( 6 q - 8 q^{5} - 12 q^{7} - 18 q^{9} - 8 q^{11} - 36 q^{15} + 24 q^{19} + 68 q^{23} + 10 q^{25} - 88 q^{35} + 208 q^{37} + 68 q^{41} + 232 q^{45} + 268 q^{47} - 62 q^{49} - 192 q^{51} - 64 q^{53} + 288 q^{55} - 360 q^{59} - 172 q^{63} + 304 q^{75} - 400 q^{77} + 238 q^{81} - 304 q^{85} - 584 q^{87} - 76 q^{89} + 208 q^{91} + 320 q^{93} - 32 q^{95} + 856 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 5.30219i 1.76740i 0.468058 + 0.883698i \(0.344954\pi\)
−0.468058 + 0.883698i \(0.655046\pi\)
\(4\) 0 0
\(5\) −4.75441 + 1.54778i −0.950881 + 0.309556i
\(6\) 0 0
\(7\) 0.206625 0.0295178 0.0147589 0.999891i \(-0.495302\pi\)
0.0147589 + 0.999891i \(0.495302\pi\)
\(8\) 0 0
\(9\) −19.1132 −2.12369
\(10\) 0 0
\(11\) −15.0176 −1.36524 −0.682619 0.730774i \(-0.739159\pi\)
−0.682619 + 0.730774i \(0.739159\pi\)
\(12\) 0 0
\(13\) −11.6999 −0.899995 −0.449998 0.893030i \(-0.648575\pi\)
−0.449998 + 0.893030i \(0.648575\pi\)
\(14\) 0 0
\(15\) −8.20662 25.2087i −0.547108 1.68058i
\(16\) 0 0
\(17\) 18.1911i 1.07007i 0.844831 + 0.535033i \(0.179701\pi\)
−0.844831 + 0.535033i \(0.820299\pi\)
\(18\) 0 0
\(19\) −19.3999 −1.02105 −0.510523 0.859864i \(-0.670548\pi\)
−0.510523 + 0.859864i \(0.670548\pi\)
\(20\) 0 0
\(21\) 1.09556i 0.0521696i
\(22\) 0 0
\(23\) 27.2242 1.18366 0.591831 0.806062i \(-0.298405\pi\)
0.591831 + 0.806062i \(0.298405\pi\)
\(24\) 0 0
\(25\) 20.2087 14.7176i 0.808350 0.588702i
\(26\) 0 0
\(27\) 53.6220i 1.98600i
\(28\) 0 0
\(29\) 44.4175i 1.53164i 0.643056 + 0.765819i \(0.277666\pi\)
−0.643056 + 0.765819i \(0.722334\pi\)
\(30\) 0 0
\(31\) 20.3822i 0.657492i −0.944418 0.328746i \(-0.893374\pi\)
0.944418 0.328746i \(-0.106626\pi\)
\(32\) 0 0
\(33\) 79.6262i 2.41292i
\(34\) 0 0
\(35\) −0.982377 + 0.319810i −0.0280679 + 0.00913742i
\(36\) 0 0
\(37\) 18.1089 0.489431 0.244715 0.969595i \(-0.421305\pi\)
0.244715 + 0.969595i \(0.421305\pi\)
\(38\) 0 0
\(39\) 62.0352i 1.59065i
\(40\) 0 0
\(41\) 32.3043 0.787910 0.393955 0.919130i \(-0.371107\pi\)
0.393955 + 0.919130i \(0.371107\pi\)
\(42\) 0 0
\(43\) 4.06244i 0.0944753i 0.998884 + 0.0472377i \(0.0150418\pi\)
−0.998884 + 0.0472377i \(0.984958\pi\)
\(44\) 0 0
\(45\) 90.8718 29.5830i 2.01937 0.657401i
\(46\) 0 0
\(47\) 5.37588 0.114380 0.0571902 0.998363i \(-0.481786\pi\)
0.0571902 + 0.998363i \(0.481786\pi\)
\(48\) 0 0
\(49\) −48.9573 −0.999129
\(50\) 0 0
\(51\) −96.4527 −1.89123
\(52\) 0 0
\(53\) 79.1703 1.49378 0.746890 0.664948i \(-0.231546\pi\)
0.746890 + 0.664948i \(0.231546\pi\)
\(54\) 0 0
\(55\) 71.3999 23.2440i 1.29818 0.422618i
\(56\) 0 0
\(57\) 102.862i 1.80459i
\(58\) 0 0
\(59\) −83.3999 −1.41356 −0.706779 0.707435i \(-0.749852\pi\)
−0.706779 + 0.707435i \(0.749852\pi\)
\(60\) 0 0
\(61\) 36.7486i 0.602435i −0.953555 0.301218i \(-0.902607\pi\)
0.953555 0.301218i \(-0.0973931\pi\)
\(62\) 0 0
\(63\) −3.94925 −0.0626866
\(64\) 0 0
\(65\) 55.6262 18.1089i 0.855788 0.278599i
\(66\) 0 0
\(67\) 4.51518i 0.0673907i −0.999432 0.0336954i \(-0.989272\pi\)
0.999432 0.0336954i \(-0.0107276\pi\)
\(68\) 0 0
\(69\) 144.348i 2.09200i
\(70\) 0 0
\(71\) 41.6530i 0.586662i 0.956011 + 0.293331i \(0.0947638\pi\)
−0.956011 + 0.293331i \(0.905236\pi\)
\(72\) 0 0
\(73\) 41.5910i 0.569740i 0.958566 + 0.284870i \(0.0919504\pi\)
−0.958566 + 0.284870i \(0.908050\pi\)
\(74\) 0 0
\(75\) 78.0352 + 107.151i 1.04047 + 1.42867i
\(76\) 0 0
\(77\) −3.10301 −0.0402988
\(78\) 0 0
\(79\) 15.5473i 0.196801i 0.995147 + 0.0984004i \(0.0313726\pi\)
−0.995147 + 0.0984004i \(0.968627\pi\)
\(80\) 0 0
\(81\) 112.295 1.38636
\(82\) 0 0
\(83\) 50.9862i 0.614291i −0.951663 0.307146i \(-0.900626\pi\)
0.951663 0.307146i \(-0.0993739\pi\)
\(84\) 0 0
\(85\) −28.1559 86.4880i −0.331246 1.01751i
\(86\) 0 0
\(87\) −235.510 −2.70701
\(88\) 0 0
\(89\) −10.8885 −0.122343 −0.0611713 0.998127i \(-0.519484\pi\)
−0.0611713 + 0.998127i \(0.519484\pi\)
\(90\) 0 0
\(91\) −2.41749 −0.0265659
\(92\) 0 0
\(93\) 108.070 1.16205
\(94\) 0 0
\(95\) 92.2349 30.0268i 0.970893 0.316071i
\(96\) 0 0
\(97\) 12.1559i 0.125318i 0.998035 + 0.0626592i \(0.0199581\pi\)
−0.998035 + 0.0626592i \(0.980042\pi\)
\(98\) 0 0
\(99\) 287.035 2.89934
\(100\) 0 0
\(101\) 127.723i 1.26459i −0.774728 0.632294i \(-0.782113\pi\)
0.774728 0.632294i \(-0.217887\pi\)
\(102\) 0 0
\(103\) −4.77575 −0.0463665 −0.0231833 0.999731i \(-0.507380\pi\)
−0.0231833 + 0.999731i \(0.507380\pi\)
\(104\) 0 0
\(105\) −1.69569 5.20875i −0.0161494 0.0496071i
\(106\) 0 0
\(107\) 107.213i 1.00199i −0.865449 0.500997i \(-0.832967\pi\)
0.865449 0.500997i \(-0.167033\pi\)
\(108\) 0 0
\(109\) 53.6689i 0.492376i 0.969222 + 0.246188i \(0.0791780\pi\)
−0.969222 + 0.246188i \(0.920822\pi\)
\(110\) 0 0
\(111\) 96.0170i 0.865018i
\(112\) 0 0
\(113\) 20.5063i 0.181471i −0.995875 0.0907356i \(-0.971078\pi\)
0.995875 0.0907356i \(-0.0289218\pi\)
\(114\) 0 0
\(115\) −129.435 + 42.1372i −1.12552 + 0.366410i
\(116\) 0 0
\(117\) 223.623 1.91131
\(118\) 0 0
\(119\) 3.75873i 0.0315860i
\(120\) 0 0
\(121\) 104.529 0.863876
\(122\) 0 0
\(123\) 171.283i 1.39255i
\(124\) 0 0
\(125\) −73.3010 + 101.252i −0.586408 + 0.810016i
\(126\) 0 0
\(127\) −138.477 −1.09037 −0.545184 0.838316i \(-0.683540\pi\)
−0.545184 + 0.838316i \(0.683540\pi\)
\(128\) 0 0
\(129\) −21.5398 −0.166975
\(130\) 0 0
\(131\) 219.105 1.67256 0.836279 0.548304i \(-0.184726\pi\)
0.836279 + 0.548304i \(0.184726\pi\)
\(132\) 0 0
\(133\) −4.00849 −0.0301390
\(134\) 0 0
\(135\) 82.9951 + 254.941i 0.614779 + 1.88845i
\(136\) 0 0
\(137\) 59.7821i 0.436366i −0.975908 0.218183i \(-0.929987\pi\)
0.975908 0.218183i \(-0.0700129\pi\)
\(138\) 0 0
\(139\) 26.6524 0.191744 0.0958718 0.995394i \(-0.469436\pi\)
0.0958718 + 0.995394i \(0.469436\pi\)
\(140\) 0 0
\(141\) 28.5039i 0.202155i
\(142\) 0 0
\(143\) 175.705 1.22871
\(144\) 0 0
\(145\) −68.7486 211.179i −0.474128 1.45641i
\(146\) 0 0
\(147\) 259.581i 1.76586i
\(148\) 0 0
\(149\) 143.463i 0.962838i −0.876490 0.481419i \(-0.840121\pi\)
0.876490 0.481419i \(-0.159879\pi\)
\(150\) 0 0
\(151\) 83.4937i 0.552939i 0.961023 + 0.276469i \(0.0891644\pi\)
−0.961023 + 0.276469i \(0.910836\pi\)
\(152\) 0 0
\(153\) 347.690i 2.27249i
\(154\) 0 0
\(155\) 31.5473 + 96.9055i 0.203531 + 0.625197i
\(156\) 0 0
\(157\) 169.673 1.08072 0.540360 0.841434i \(-0.318288\pi\)
0.540360 + 0.841434i \(0.318288\pi\)
\(158\) 0 0
\(159\) 419.776i 2.64010i
\(160\) 0 0
\(161\) 5.62520 0.0349391
\(162\) 0 0
\(163\) 275.478i 1.69005i 0.534726 + 0.845026i \(0.320415\pi\)
−0.534726 + 0.845026i \(0.679585\pi\)
\(164\) 0 0
\(165\) 123.244 + 378.575i 0.746933 + 2.29440i
\(166\) 0 0
\(167\) −132.481 −0.793299 −0.396650 0.917970i \(-0.629827\pi\)
−0.396650 + 0.917970i \(0.629827\pi\)
\(168\) 0 0
\(169\) −32.1115 −0.190009
\(170\) 0 0
\(171\) 370.793 2.16838
\(172\) 0 0
\(173\) 272.614 1.57580 0.787901 0.615801i \(-0.211168\pi\)
0.787901 + 0.615801i \(0.211168\pi\)
\(174\) 0 0
\(175\) 4.17562 3.04101i 0.0238607 0.0173772i
\(176\) 0 0
\(177\) 442.202i 2.49831i
\(178\) 0 0
\(179\) −157.523 −0.880014 −0.440007 0.897994i \(-0.645024\pi\)
−0.440007 + 0.897994i \(0.645024\pi\)
\(180\) 0 0
\(181\) 335.063i 1.85118i 0.378529 + 0.925590i \(0.376430\pi\)
−0.378529 + 0.925590i \(0.623570\pi\)
\(182\) 0 0
\(183\) 194.848 1.06474
\(184\) 0 0
\(185\) −86.0972 + 28.0287i −0.465391 + 0.151506i
\(186\) 0 0
\(187\) 273.187i 1.46090i
\(188\) 0 0
\(189\) 11.0796i 0.0586223i
\(190\) 0 0
\(191\) 298.575i 1.56322i −0.623766 0.781611i \(-0.714398\pi\)
0.623766 0.781611i \(-0.285602\pi\)
\(192\) 0 0
\(193\) 191.915i 0.994376i 0.867643 + 0.497188i \(0.165634\pi\)
−0.867643 + 0.497188i \(0.834366\pi\)
\(194\) 0 0
\(195\) 96.0170 + 294.941i 0.492395 + 1.51252i
\(196\) 0 0
\(197\) −59.2472 −0.300747 −0.150374 0.988629i \(-0.548048\pi\)
−0.150374 + 0.988629i \(0.548048\pi\)
\(198\) 0 0
\(199\) 309.100i 1.55327i −0.629953 0.776633i \(-0.716926\pi\)
0.629953 0.776633i \(-0.283074\pi\)
\(200\) 0 0
\(201\) 23.9403 0.119106
\(202\) 0 0
\(203\) 9.17775i 0.0452106i
\(204\) 0 0
\(205\) −153.588 + 50.0000i −0.749209 + 0.243902i
\(206\) 0 0
\(207\) −520.342 −2.51373
\(208\) 0 0
\(209\) 291.340 1.39397
\(210\) 0 0
\(211\) −205.693 −0.974850 −0.487425 0.873165i \(-0.662064\pi\)
−0.487425 + 0.873165i \(0.662064\pi\)
\(212\) 0 0
\(213\) −220.852 −1.03686
\(214\) 0 0
\(215\) −6.28777 19.3145i −0.0292454 0.0898348i
\(216\) 0 0
\(217\) 4.21147i 0.0194077i
\(218\) 0 0
\(219\) −220.523 −1.00696
\(220\) 0 0
\(221\) 212.835i 0.963054i
\(222\) 0 0
\(223\) −228.723 −1.02567 −0.512833 0.858488i \(-0.671404\pi\)
−0.512833 + 0.858488i \(0.671404\pi\)
\(224\) 0 0
\(225\) −386.254 + 281.299i −1.71668 + 1.25022i
\(226\) 0 0
\(227\) 282.403i 1.24407i −0.782991 0.622033i \(-0.786307\pi\)
0.782991 0.622033i \(-0.213693\pi\)
\(228\) 0 0
\(229\) 49.2525i 0.215076i −0.994201 0.107538i \(-0.965703\pi\)
0.994201 0.107538i \(-0.0342968\pi\)
\(230\) 0 0
\(231\) 16.4527i 0.0712240i
\(232\) 0 0
\(233\) 124.273i 0.533363i 0.963785 + 0.266681i \(0.0859271\pi\)
−0.963785 + 0.266681i \(0.914073\pi\)
\(234\) 0 0
\(235\) −25.5591 + 8.32069i −0.108762 + 0.0354072i
\(236\) 0 0
\(237\) −82.4345 −0.347825
\(238\) 0 0
\(239\) 80.4527i 0.336622i −0.985734 0.168311i \(-0.946169\pi\)
0.985734 0.168311i \(-0.0538313\pi\)
\(240\) 0 0
\(241\) −1.20979 −0.00501988 −0.00250994 0.999997i \(-0.500799\pi\)
−0.00250994 + 0.999997i \(0.500799\pi\)
\(242\) 0 0
\(243\) 112.812i 0.464247i
\(244\) 0 0
\(245\) 232.763 75.7752i 0.950053 0.309287i
\(246\) 0 0
\(247\) 226.977 0.918936
\(248\) 0 0
\(249\) 270.338 1.08570
\(250\) 0 0
\(251\) −211.853 −0.844034 −0.422017 0.906588i \(-0.638678\pi\)
−0.422017 + 0.906588i \(0.638678\pi\)
\(252\) 0 0
\(253\) −408.843 −1.61598
\(254\) 0 0
\(255\) 458.575 149.288i 1.79834 0.585442i
\(256\) 0 0
\(257\) 182.646i 0.710685i 0.934736 + 0.355342i \(0.115636\pi\)
−0.934736 + 0.355342i \(0.884364\pi\)
\(258\) 0 0
\(259\) 3.74175 0.0144469
\(260\) 0 0
\(261\) 848.960i 3.25272i
\(262\) 0 0
\(263\) −74.6636 −0.283892 −0.141946 0.989874i \(-0.545336\pi\)
−0.141946 + 0.989874i \(0.545336\pi\)
\(264\) 0 0
\(265\) −376.408 + 122.538i −1.42041 + 0.462409i
\(266\) 0 0
\(267\) 57.7329i 0.216228i
\(268\) 0 0
\(269\) 184.089i 0.684344i −0.939637 0.342172i \(-0.888837\pi\)
0.939637 0.342172i \(-0.111163\pi\)
\(270\) 0 0
\(271\) 234.746i 0.866222i −0.901341 0.433111i \(-0.857416\pi\)
0.901341 0.433111i \(-0.142584\pi\)
\(272\) 0 0
\(273\) 12.8180i 0.0469524i
\(274\) 0 0
\(275\) −303.487 + 221.023i −1.10359 + 0.803719i
\(276\) 0 0
\(277\) −452.208 −1.63252 −0.816260 0.577684i \(-0.803957\pi\)
−0.816260 + 0.577684i \(0.803957\pi\)
\(278\) 0 0
\(279\) 389.570i 1.39631i
\(280\) 0 0
\(281\) 196.110 0.697900 0.348950 0.937141i \(-0.386538\pi\)
0.348950 + 0.937141i \(0.386538\pi\)
\(282\) 0 0
\(283\) 418.449i 1.47862i −0.673366 0.739309i \(-0.735152\pi\)
0.673366 0.739309i \(-0.264848\pi\)
\(284\) 0 0
\(285\) 159.207 + 489.046i 0.558623 + 1.71595i
\(286\) 0 0
\(287\) 6.67486 0.0232574
\(288\) 0 0
\(289\) −41.9170 −0.145042
\(290\) 0 0
\(291\) −64.4527 −0.221487
\(292\) 0 0
\(293\) 286.666 0.978383 0.489191 0.872176i \(-0.337292\pi\)
0.489191 + 0.872176i \(0.337292\pi\)
\(294\) 0 0
\(295\) 396.517 129.085i 1.34412 0.437575i
\(296\) 0 0
\(297\) 805.275i 2.71136i
\(298\) 0 0
\(299\) −318.522 −1.06529
\(300\) 0 0
\(301\) 0.839400i 0.00278870i
\(302\) 0 0
\(303\) 677.214 2.23503
\(304\) 0 0
\(305\) 56.8787 + 174.718i 0.186488 + 0.572844i
\(306\) 0 0
\(307\) 261.715i 0.852493i −0.904607 0.426247i \(-0.859836\pi\)
0.904607 0.426247i \(-0.140164\pi\)
\(308\) 0 0
\(309\) 25.3219i 0.0819480i
\(310\) 0 0
\(311\) 578.904i 1.86143i −0.365747 0.930714i \(-0.619187\pi\)
0.365747 0.930714i \(-0.380813\pi\)
\(312\) 0 0
\(313\) 99.3124i 0.317292i −0.987336 0.158646i \(-0.949287\pi\)
0.987336 0.158646i \(-0.0507129\pi\)
\(314\) 0 0
\(315\) 18.7764 6.11258i 0.0596075 0.0194050i
\(316\) 0 0
\(317\) 191.623 0.604489 0.302245 0.953230i \(-0.402264\pi\)
0.302245 + 0.953230i \(0.402264\pi\)
\(318\) 0 0
\(319\) 667.045i 2.09105i
\(320\) 0 0
\(321\) 568.466 1.77092
\(322\) 0 0
\(323\) 352.905i 1.09259i
\(324\) 0 0
\(325\) −236.441 + 172.194i −0.727511 + 0.529829i
\(326\) 0 0
\(327\) −284.563 −0.870222
\(328\) 0 0
\(329\) 1.11079 0.00337626
\(330\) 0 0
\(331\) −530.187 −1.60177 −0.800886 0.598816i \(-0.795638\pi\)
−0.800886 + 0.598816i \(0.795638\pi\)
\(332\) 0 0
\(333\) −346.120 −1.03940
\(334\) 0 0
\(335\) 6.98851 + 21.4670i 0.0208612 + 0.0640806i
\(336\) 0 0
\(337\) 487.427i 1.44637i 0.690653 + 0.723186i \(0.257323\pi\)
−0.690653 + 0.723186i \(0.742677\pi\)
\(338\) 0 0
\(339\) 108.728 0.320731
\(340\) 0 0
\(341\) 306.093i 0.897633i
\(342\) 0 0
\(343\) −20.2404 −0.0590099
\(344\) 0 0
\(345\) −223.419 686.289i −0.647592 1.98924i
\(346\) 0 0
\(347\) 310.497i 0.894804i 0.894333 + 0.447402i \(0.147651\pi\)
−0.894333 + 0.447402i \(0.852349\pi\)
\(348\) 0 0
\(349\) 253.004i 0.724941i −0.931995 0.362471i \(-0.881933\pi\)
0.931995 0.362471i \(-0.118067\pi\)
\(350\) 0 0
\(351\) 627.374i 1.78739i
\(352\) 0 0
\(353\) 322.639i 0.913992i −0.889469 0.456996i \(-0.848925\pi\)
0.889469 0.456996i \(-0.151075\pi\)
\(354\) 0 0
\(355\) −64.4697 198.035i −0.181605 0.557846i
\(356\) 0 0
\(357\) −19.9295 −0.0558250
\(358\) 0 0
\(359\) 254.975i 0.710236i −0.934822 0.355118i \(-0.884441\pi\)
0.934822 0.355118i \(-0.115559\pi\)
\(360\) 0 0
\(361\) 15.3550 0.0425347
\(362\) 0 0
\(363\) 554.232i 1.52681i
\(364\) 0 0
\(365\) −64.3738 197.740i −0.176366 0.541755i
\(366\) 0 0
\(367\) −207.935 −0.566581 −0.283291 0.959034i \(-0.591426\pi\)
−0.283291 + 0.959034i \(0.591426\pi\)
\(368\) 0 0
\(369\) −617.438 −1.67327
\(370\) 0 0
\(371\) 16.3585 0.0440931
\(372\) 0 0
\(373\) 203.826 0.546450 0.273225 0.961950i \(-0.411910\pi\)
0.273225 + 0.961950i \(0.411910\pi\)
\(374\) 0 0
\(375\) −536.857 388.656i −1.43162 1.03642i
\(376\) 0 0
\(377\) 519.682i 1.37847i
\(378\) 0 0
\(379\) 454.099 1.19815 0.599076 0.800692i \(-0.295535\pi\)
0.599076 + 0.800692i \(0.295535\pi\)
\(380\) 0 0
\(381\) 734.229i 1.92711i
\(382\) 0 0
\(383\) −541.569 −1.41402 −0.707009 0.707205i \(-0.749956\pi\)
−0.707009 + 0.707205i \(0.749956\pi\)
\(384\) 0 0
\(385\) 14.7530 4.80278i 0.0383194 0.0124748i
\(386\) 0 0
\(387\) 77.6462i 0.200636i
\(388\) 0 0
\(389\) 423.431i 1.08851i 0.838919 + 0.544256i \(0.183188\pi\)
−0.838919 + 0.544256i \(0.816812\pi\)
\(390\) 0 0
\(391\) 495.240i 1.26660i
\(392\) 0 0
\(393\) 1161.74i 2.95607i
\(394\) 0 0
\(395\) −24.0638 73.9180i −0.0609209 0.187134i
\(396\) 0 0
\(397\) −11.7772 −0.0296654 −0.0148327 0.999890i \(-0.504722\pi\)
−0.0148327 + 0.999890i \(0.504722\pi\)
\(398\) 0 0
\(399\) 21.2538i 0.0532676i
\(400\) 0 0
\(401\) 127.442 0.317809 0.158905 0.987294i \(-0.449204\pi\)
0.158905 + 0.987294i \(0.449204\pi\)
\(402\) 0 0
\(403\) 238.471i 0.591739i
\(404\) 0 0
\(405\) −533.897 + 173.808i −1.31826 + 0.429156i
\(406\) 0 0
\(407\) −271.953 −0.668190
\(408\) 0 0
\(409\) −608.012 −1.48658 −0.743290 0.668969i \(-0.766736\pi\)
−0.743290 + 0.668969i \(0.766736\pi\)
\(410\) 0 0
\(411\) 316.976 0.771231
\(412\) 0 0
\(413\) −17.2325 −0.0417251
\(414\) 0 0
\(415\) 78.9155 + 242.409i 0.190158 + 0.584118i
\(416\) 0 0
\(417\) 141.316i 0.338887i
\(418\) 0 0
\(419\) −565.630 −1.34995 −0.674976 0.737840i \(-0.735846\pi\)
−0.674976 + 0.737840i \(0.735846\pi\)
\(420\) 0 0
\(421\) 711.356i 1.68968i −0.535018 0.844841i \(-0.679695\pi\)
0.535018 0.844841i \(-0.320305\pi\)
\(422\) 0 0
\(423\) −102.750 −0.242908
\(424\) 0 0
\(425\) 267.729 + 367.620i 0.629950 + 0.864988i
\(426\) 0 0
\(427\) 7.59316i 0.0177826i
\(428\) 0 0
\(429\) 931.622i 2.17161i
\(430\) 0 0
\(431\) 309.254i 0.717526i 0.933429 + 0.358763i \(0.116801\pi\)
−0.933429 + 0.358763i \(0.883199\pi\)
\(432\) 0 0
\(433\) 187.374i 0.432735i 0.976312 + 0.216368i \(0.0694210\pi\)
−0.976312 + 0.216368i \(0.930579\pi\)
\(434\) 0 0
\(435\) 1119.71 364.518i 2.57404 0.837972i
\(436\) 0 0
\(437\) −528.147 −1.20857
\(438\) 0 0
\(439\) 289.657i 0.659811i −0.944014 0.329906i \(-0.892983\pi\)
0.944014 0.329906i \(-0.107017\pi\)
\(440\) 0 0
\(441\) 935.730 2.12184
\(442\) 0 0
\(443\) 295.516i 0.667079i 0.942736 + 0.333539i \(0.108243\pi\)
−0.942736 + 0.333539i \(0.891757\pi\)
\(444\) 0 0
\(445\) 51.7683 16.8530i 0.116333 0.0378719i
\(446\) 0 0
\(447\) 760.667 1.70172
\(448\) 0 0
\(449\) −604.409 −1.34612 −0.673061 0.739587i \(-0.735021\pi\)
−0.673061 + 0.739587i \(0.735021\pi\)
\(450\) 0 0
\(451\) −485.134 −1.07569
\(452\) 0 0
\(453\) −442.699 −0.977262
\(454\) 0 0
\(455\) 11.4937 3.74175i 0.0252610 0.00822363i
\(456\) 0 0
\(457\) 392.507i 0.858877i 0.903096 + 0.429438i \(0.141288\pi\)
−0.903096 + 0.429438i \(0.858712\pi\)
\(458\) 0 0
\(459\) 975.444 2.12515
\(460\) 0 0
\(461\) 400.277i 0.868279i −0.900846 0.434139i \(-0.857053\pi\)
0.900846 0.434139i \(-0.142947\pi\)
\(462\) 0 0
\(463\) 732.679 1.58246 0.791230 0.611518i \(-0.209441\pi\)
0.791230 + 0.611518i \(0.209441\pi\)
\(464\) 0 0
\(465\) −513.811 + 167.269i −1.10497 + 0.359719i
\(466\) 0 0
\(467\) 592.126i 1.26794i 0.773360 + 0.633968i \(0.218575\pi\)
−0.773360 + 0.633968i \(0.781425\pi\)
\(468\) 0 0
\(469\) 0.932947i 0.00198923i
\(470\) 0 0
\(471\) 899.639i 1.91006i
\(472\) 0 0
\(473\) 61.0082i 0.128981i
\(474\) 0 0
\(475\) −392.047 + 285.519i −0.825362 + 0.601092i
\(476\) 0 0
\(477\) −1513.20 −3.17232
\(478\) 0 0
\(479\) 309.151i 0.645409i 0.946500 + 0.322705i \(0.104592\pi\)
−0.946500 + 0.322705i \(0.895408\pi\)
\(480\) 0 0
\(481\) −211.873 −0.440485
\(482\) 0 0
\(483\) 29.8259i 0.0617513i
\(484\) 0 0
\(485\) −18.8146 57.7940i −0.0387931 0.119163i
\(486\) 0 0
\(487\) 570.769 1.17201 0.586005 0.810307i \(-0.300700\pi\)
0.586005 + 0.810307i \(0.300700\pi\)
\(488\) 0 0
\(489\) −1460.64 −2.98699
\(490\) 0 0
\(491\) −301.659 −0.614378 −0.307189 0.951649i \(-0.599388\pi\)
−0.307189 + 0.951649i \(0.599388\pi\)
\(492\) 0 0
\(493\) −808.004 −1.63895
\(494\) 0 0
\(495\) −1364.68 + 444.267i −2.75693 + 0.897509i
\(496\) 0 0
\(497\) 8.60653i 0.0173170i
\(498\) 0 0
\(499\) 517.758 1.03759 0.518795 0.854898i \(-0.326381\pi\)
0.518795 + 0.854898i \(0.326381\pi\)
\(500\) 0 0
\(501\) 702.439i 1.40207i
\(502\) 0 0
\(503\) 406.671 0.808491 0.404246 0.914650i \(-0.367534\pi\)
0.404246 + 0.914650i \(0.367534\pi\)
\(504\) 0 0
\(505\) 197.688 + 607.249i 0.391461 + 1.20247i
\(506\) 0 0
\(507\) 170.261i 0.335821i
\(508\) 0 0
\(509\) 627.097i 1.23202i 0.787739 + 0.616009i \(0.211252\pi\)
−0.787739 + 0.616009i \(0.788748\pi\)
\(510\) 0 0
\(511\) 8.59372i 0.0168175i
\(512\) 0 0
\(513\) 1040.26i 2.02780i
\(514\) 0 0
\(515\) 22.7059 7.39182i 0.0440891 0.0143530i
\(516\) 0 0
\(517\) −80.7330 −0.156157
\(518\) 0 0
\(519\) 1445.45i 2.78507i
\(520\) 0 0
\(521\) 111.743 0.214478 0.107239 0.994233i \(-0.465799\pi\)
0.107239 + 0.994233i \(0.465799\pi\)
\(522\) 0 0
\(523\) 769.813i 1.47192i −0.677027 0.735959i \(-0.736732\pi\)
0.677027 0.735959i \(-0.263268\pi\)
\(524\) 0 0
\(525\) 16.1240 + 22.1399i 0.0307124 + 0.0421713i
\(526\) 0 0
\(527\) 370.776 0.703560
\(528\) 0 0
\(529\) 212.160 0.401058
\(530\) 0 0
\(531\) 1594.04 3.00195
\(532\) 0 0
\(533\) −377.958 −0.709115
\(534\) 0 0
\(535\) 165.943 + 509.736i 0.310174 + 0.952778i
\(536\) 0 0
\(537\) 835.214i 1.55533i
\(538\) 0 0
\(539\) 735.222 1.36405
\(540\) 0 0
\(541\) 225.558i 0.416927i −0.978030 0.208463i \(-0.933154\pi\)
0.978030 0.208463i \(-0.0668462\pi\)
\(542\) 0 0
\(543\) −1776.57 −3.27177
\(544\) 0 0
\(545\) −83.0678 255.164i −0.152418 0.468191i
\(546\) 0 0
\(547\) 882.346i 1.61306i 0.591190 + 0.806532i \(0.298658\pi\)
−0.591190 + 0.806532i \(0.701342\pi\)
\(548\) 0 0
\(549\) 702.382i 1.27938i
\(550\) 0 0
\(551\) 861.694i 1.56387i
\(552\) 0 0
\(553\) 3.21245i 0.00580913i
\(554\) 0 0
\(555\) −148.613 456.504i −0.267772 0.822529i
\(556\) 0 0
\(557\) 303.119 0.544199 0.272100 0.962269i \(-0.412282\pi\)
0.272100 + 0.962269i \(0.412282\pi\)
\(558\) 0 0
\(559\) 47.5303i 0.0850273i
\(560\) 0 0
\(561\) 1448.49 2.58198
\(562\) 0 0
\(563\) 344.003i 0.611017i −0.952189 0.305508i \(-0.901174\pi\)
0.952189 0.305508i \(-0.0988264\pi\)
\(564\) 0 0
\(565\) 31.7392 + 97.4950i 0.0561756 + 0.172558i
\(566\) 0 0
\(567\) 23.2029 0.0409223
\(568\) 0 0
\(569\) 228.925 0.402329 0.201165 0.979557i \(-0.435527\pi\)
0.201165 + 0.979557i \(0.435527\pi\)
\(570\) 0 0
\(571\) 371.169 0.650033 0.325017 0.945708i \(-0.394630\pi\)
0.325017 + 0.945708i \(0.394630\pi\)
\(572\) 0 0
\(573\) 1583.10 2.76283
\(574\) 0 0
\(575\) 550.168 400.674i 0.956814 0.696825i
\(576\) 0 0
\(577\) 580.289i 1.00570i 0.864374 + 0.502850i \(0.167715\pi\)
−0.864374 + 0.502850i \(0.832285\pi\)
\(578\) 0 0
\(579\) −1017.57 −1.75746
\(580\) 0 0
\(581\) 10.5350i 0.0181325i
\(582\) 0 0
\(583\) −1188.95 −2.03936
\(584\) 0 0
\(585\) −1063.19 + 346.120i −1.81743 + 0.591657i
\(586\) 0 0
\(587\) 65.0801i 0.110869i −0.998462 0.0554345i \(-0.982346\pi\)
0.998462 0.0554345i \(-0.0176544\pi\)
\(588\) 0 0
\(589\) 395.413i 0.671329i
\(590\) 0 0
\(591\) 314.140i 0.531539i
\(592\) 0 0
\(593\) 1002.90i 1.69124i −0.533787 0.845619i \(-0.679232\pi\)
0.533787 0.845619i \(-0.320768\pi\)
\(594\) 0 0
\(595\) −5.81770 17.8705i −0.00977764 0.0300345i
\(596\) 0 0
\(597\) 1638.91 2.74524
\(598\) 0 0
\(599\) 888.567i 1.48342i 0.670722 + 0.741709i \(0.265984\pi\)
−0.670722 + 0.741709i \(0.734016\pi\)
\(600\) 0 0
\(601\) −132.065 −0.219742 −0.109871 0.993946i \(-0.535044\pi\)
−0.109871 + 0.993946i \(0.535044\pi\)
\(602\) 0 0
\(603\) 86.2995i 0.143117i
\(604\) 0 0
\(605\) −496.973 + 161.788i −0.821443 + 0.267418i
\(606\) 0 0
\(607\) 700.090 1.15336 0.576680 0.816970i \(-0.304348\pi\)
0.576680 + 0.816970i \(0.304348\pi\)
\(608\) 0 0
\(609\) −48.6621 −0.0799050
\(610\) 0 0
\(611\) −62.8975 −0.102942
\(612\) 0 0
\(613\) −727.420 −1.18666 −0.593328 0.804961i \(-0.702186\pi\)
−0.593328 + 0.804961i \(0.702186\pi\)
\(614\) 0 0
\(615\) −265.109 814.351i −0.431072 1.32415i
\(616\) 0 0
\(617\) 99.7137i 0.161611i 0.996730 + 0.0808053i \(0.0257492\pi\)
−0.996730 + 0.0808053i \(0.974251\pi\)
\(618\) 0 0
\(619\) −721.349 −1.16535 −0.582673 0.812707i \(-0.697993\pi\)
−0.582673 + 0.812707i \(0.697993\pi\)
\(620\) 0 0
\(621\) 1459.82i 2.35075i
\(622\) 0 0
\(623\) −2.24983 −0.00361129
\(624\) 0 0
\(625\) 191.787 594.847i 0.306859 0.951755i
\(626\) 0 0
\(627\) 1544.74i 2.46370i
\(628\) 0 0
\(629\) 329.422i 0.523723i
\(630\) 0 0
\(631\) 796.856i 1.26285i −0.775438 0.631423i \(-0.782471\pi\)
0.775438 0.631423i \(-0.217529\pi\)
\(632\) 0 0
\(633\) 1090.62i 1.72295i
\(634\) 0 0
\(635\) 658.375 214.332i 1.03681 0.337530i
\(636\) 0 0
\(637\) 572.797 0.899211
\(638\) 0 0
\(639\) 796.121i 1.24589i
\(640\) 0 0
\(641\) 205.013 0.319833 0.159917 0.987131i \(-0.448877\pi\)
0.159917 + 0.987131i \(0.448877\pi\)
\(642\) 0 0
\(643\) 495.044i 0.769897i −0.922938 0.384949i \(-0.874219\pi\)
0.922938 0.384949i \(-0.125781\pi\)
\(644\) 0 0
\(645\) 102.409 33.3389i 0.158774 0.0516882i
\(646\) 0 0
\(647\) 1121.84 1.73391 0.866953 0.498389i \(-0.166075\pi\)
0.866953 + 0.498389i \(0.166075\pi\)
\(648\) 0 0
\(649\) 1252.47 1.92984
\(650\) 0 0
\(651\) 22.3300 0.0343011
\(652\) 0 0
\(653\) 622.987 0.954038 0.477019 0.878893i \(-0.341717\pi\)
0.477019 + 0.878893i \(0.341717\pi\)
\(654\) 0 0
\(655\) −1041.71 + 339.127i −1.59040 + 0.517751i
\(656\) 0 0
\(657\) 794.936i 1.20995i
\(658\) 0 0
\(659\) −264.030 −0.400653 −0.200326 0.979729i \(-0.564200\pi\)
−0.200326 + 0.979729i \(0.564200\pi\)
\(660\) 0 0
\(661\) 1285.15i 1.94425i −0.234469 0.972124i \(-0.575335\pi\)
0.234469 0.972124i \(-0.424665\pi\)
\(662\) 0 0
\(663\) 1128.49 1.70210
\(664\) 0 0
\(665\) 19.0580 6.20427i 0.0286586 0.00932972i
\(666\) 0 0
\(667\) 1209.23i 1.81294i
\(668\) 0 0
\(669\) 1212.73i 1.81276i
\(670\) 0 0
\(671\) 551.876i 0.822468i
\(672\) 0 0
\(673\) 1244.16i 1.84868i 0.381565 + 0.924342i \(0.375385\pi\)
−0.381565 + 0.924342i \(0.624615\pi\)
\(674\) 0 0
\(675\) −789.185 1083.63i −1.16916 1.60538i
\(676\) 0 0
\(677\) 963.335 1.42295 0.711473 0.702713i \(-0.248028\pi\)
0.711473 + 0.702713i \(0.248028\pi\)
\(678\) 0 0
\(679\) 2.51170i 0.00369912i
\(680\) 0 0
\(681\) 1497.35 2.19876
\(682\) 0 0
\(683\) 770.819i 1.12858i −0.825577 0.564289i \(-0.809150\pi\)
0.825577 0.564289i \(-0.190850\pi\)
\(684\) 0 0
\(685\) 92.5296 + 284.228i 0.135080 + 0.414932i
\(686\) 0 0
\(687\) 261.146 0.380125
\(688\) 0 0
\(689\) −926.287 −1.34439
\(690\) 0 0
\(691\) −408.765 −0.591555 −0.295778 0.955257i \(-0.595579\pi\)
−0.295778 + 0.955257i \(0.595579\pi\)
\(692\) 0 0
\(693\) 59.3084 0.0855821
\(694\) 0 0
\(695\) −126.716 + 41.2520i −0.182325 + 0.0593554i
\(696\) 0 0
\(697\) 587.652i 0.843116i
\(698\) 0 0
\(699\) −658.921 −0.942663
\(700\) 0 0
\(701\) 1335.62i 1.90530i 0.304068 + 0.952650i \(0.401655\pi\)
−0.304068 + 0.952650i \(0.598345\pi\)
\(702\) 0 0
\(703\) −351.311 −0.499731
\(704\) 0 0
\(705\) −44.1178 135.519i −0.0625785 0.192226i
\(706\) 0 0
\(707\) 26.3908i 0.0373279i
\(708\) 0 0
\(709\) 362.956i 0.511927i −0.966686 0.255964i \(-0.917607\pi\)
0.966686 0.255964i \(-0.0823927\pi\)
\(710\) 0 0
\(711\) 297.158i 0.417943i
\(712\) 0 0
\(713\) 554.891i 0.778249i
\(714\) 0 0
\(715\) −835.374 + 271.953i −1.16836 + 0.380354i
\(716\) 0 0
\(717\) 426.575 0.594945
\(718\) 0 0
\(719\) 648.098i 0.901388i −0.892679 0.450694i \(-0.851177\pi\)
0.892679 0.450694i \(-0.148823\pi\)
\(720\) 0 0
\(721\) −0.986788 −0.00136864
\(722\) 0 0
\(723\) 6.41453i 0.00887211i
\(724\) 0 0
\(725\) 653.717 + 897.622i 0.901679 + 1.23810i
\(726\) 0 0
\(727\) −431.123 −0.593017 −0.296508 0.955030i \(-0.595822\pi\)
−0.296508 + 0.955030i \(0.595822\pi\)
\(728\) 0 0
\(729\) 412.506 0.565852
\(730\) 0 0
\(731\) −73.9003 −0.101095
\(732\) 0 0
\(733\) −1464.87 −1.99846 −0.999231 0.0392126i \(-0.987515\pi\)
−0.999231 + 0.0392126i \(0.987515\pi\)
\(734\) 0 0
\(735\) 401.774 + 1234.15i 0.546632 + 1.67912i
\(736\) 0 0
\(737\) 67.8073i 0.0920044i
\(738\) 0 0
\(739\) −99.1951 −0.134229 −0.0671144 0.997745i \(-0.521379\pi\)
−0.0671144 + 0.997745i \(0.521379\pi\)
\(740\) 0 0
\(741\) 1203.48i 1.62412i
\(742\) 0 0
\(743\) −602.719 −0.811196 −0.405598 0.914052i \(-0.632937\pi\)
−0.405598 + 0.914052i \(0.632937\pi\)
\(744\) 0 0
\(745\) 222.049 + 682.081i 0.298053 + 0.915545i
\(746\) 0 0
\(747\) 974.508i 1.30456i
\(748\) 0 0
\(749\) 22.1529i 0.0295767i
\(750\) 0 0
\(751\) 541.472i 0.721001i −0.932759 0.360501i \(-0.882606\pi\)
0.932759 0.360501i \(-0.117394\pi\)
\(752\) 0 0
\(753\) 1123.28i 1.49174i
\(754\) 0 0
\(755\) −129.230 396.963i −0.171166 0.525779i
\(756\) 0 0
\(757\) −1092.79 −1.44358 −0.721789 0.692113i \(-0.756680\pi\)
−0.721789 + 0.692113i \(0.756680\pi\)
\(758\) 0 0
\(759\) 2167.76i 2.85608i
\(760\) 0 0
\(761\) 18.7706 0.0246657 0.0123328 0.999924i \(-0.496074\pi\)
0.0123328 + 0.999924i \(0.496074\pi\)
\(762\) 0 0
\(763\) 11.0893i 0.0145338i
\(764\) 0 0
\(765\) 538.149 + 1653.06i 0.703462 + 2.16086i
\(766\) 0 0
\(767\) 975.773 1.27219
\(768\) 0 0
\(769\) 39.0830 0.0508231 0.0254116 0.999677i \(-0.491910\pi\)
0.0254116 + 0.999677i \(0.491910\pi\)
\(770\) 0 0
\(771\) −968.423 −1.25606
\(772\) 0 0
\(773\) 31.2171 0.0403843 0.0201922 0.999796i \(-0.493572\pi\)
0.0201922 + 0.999796i \(0.493572\pi\)
\(774\) 0 0
\(775\) −299.977 411.900i −0.387067 0.531484i
\(776\) 0 0
\(777\) 19.8395i 0.0255334i
\(778\) 0 0
\(779\) −626.699 −0.804492
\(780\) 0 0
\(781\) 625.529i 0.800933i
\(782\) 0 0
\(783\) 2381.75 3.04183
\(784\) 0 0
\(785\) −806.695 + 262.617i −1.02764 + 0.334544i
\(786\) 0 0
\(787\) 988.563i 1.25612i −0.778167 0.628058i \(-0.783850\pi\)
0.778167 0.628058i \(-0.216150\pi\)
\(788\) 0 0
\(789\) 395.880i 0.501750i
\(790\) 0 0
\(791\) 4.23710i 0.00535663i
\(792\) 0 0
\(793\) 429.956i 0.542189i
\(794\) 0 0
\(795\) −649.721 1995.78i −0.817259 2.51042i
\(796\) 0 0
\(797\) −118.920 −0.149210 −0.0746048 0.997213i \(-0.523770\pi\)
−0.0746048 + 0.997213i \(0.523770\pi\)
\(798\) 0 0
\(799\) 97.7933i 0.122395i
\(800\) 0