Properties

Label 3120.2.l.n.1249.7
Level $3120$
Weight $2$
Character 3120.1249
Analytic conductor $24.913$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.9133254306\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.57815240704.2
Defining polynomial: \(x^{8} - 2 x^{7} + 2 x^{6} + 89 x^{4} - 170 x^{3} + 162 x^{2} - 72 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.7
Root \(0.353624 + 0.353624i\) of defining polynomial
Character \(\chi\) \(=\) 3120.1249
Dual form 3120.2.l.n.1249.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +(-0.353624 - 2.20793i) q^{5} -3.09417i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +(-0.353624 - 2.20793i) q^{5} -3.09417i q^{7} -1.00000 q^{9} -5.51003 q^{11} +1.00000i q^{13} +(2.20793 - 0.353624i) q^{15} +6.21728i q^{17} +3.09417 q^{21} -4.21728i q^{23} +(-4.74990 + 1.56155i) q^{25} -1.00000i q^{27} +1.70861 q^{29} -5.51003i q^{33} +(-6.83172 + 1.09417i) q^{35} +4.02893i q^{37} -1.00000 q^{39} +0.198578 q^{41} +5.70861i q^{43} +(0.353624 + 2.20793i) q^{45} -6.41586i q^{47} -2.57391 q^{49} -6.21728 q^{51} +4.02893i q^{53} +(1.94848 + 12.1658i) q^{55} -7.53896 q^{59} +11.9259 q^{61} +3.09417i q^{63} +(2.20793 - 0.353624i) q^{65} +4.83172i q^{67} +4.21728 q^{69} -7.98977 q^{71} +9.31145i q^{73} +(-1.56155 - 4.74990i) q^{75} +17.0490i q^{77} +8.51139 q^{79} +1.00000 q^{81} +14.7926i q^{83} +(13.7273 - 2.19858i) q^{85} +1.70861i q^{87} +9.40699 q^{89} +3.09417 q^{91} -14.3026i q^{97} +5.51003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 8 q^{9} + O(q^{10}) \) \( 8 q - 2 q^{5} - 8 q^{9} - 2 q^{11} + 2 q^{15} + 14 q^{21} - 16 q^{29} + 8 q^{35} - 8 q^{39} + 14 q^{41} + 2 q^{45} - 18 q^{49} - 6 q^{51} - 10 q^{55} + 4 q^{59} + 22 q^{61} + 2 q^{65} - 10 q^{69} - 30 q^{71} + 4 q^{75} - 2 q^{79} + 8 q^{81} + 24 q^{85} - 18 q^{89} + 14 q^{91} + 2 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times\).

\(n\) \(1951\) \(2081\) \(2341\) \(2497\) \(2641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −0.353624 2.20793i −0.158145 0.987416i
\(6\) 0 0
\(7\) 3.09417i 1.16949i −0.811218 0.584744i \(-0.801195\pi\)
0.811218 0.584744i \(-0.198805\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −5.51003 −1.66134 −0.830669 0.556767i \(-0.812042\pi\)
−0.830669 + 0.556767i \(0.812042\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 2.20793 0.353624i 0.570085 0.0913053i
\(16\) 0 0
\(17\) 6.21728i 1.50791i 0.656925 + 0.753956i \(0.271857\pi\)
−0.656925 + 0.753956i \(0.728143\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 3.09417 0.675204
\(22\) 0 0
\(23\) 4.21728i 0.879364i −0.898154 0.439682i \(-0.855091\pi\)
0.898154 0.439682i \(-0.144909\pi\)
\(24\) 0 0
\(25\) −4.74990 + 1.56155i −0.949980 + 0.312311i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 1.70861 0.317281 0.158640 0.987336i \(-0.449289\pi\)
0.158640 + 0.987336i \(0.449289\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 5.51003i 0.959173i
\(34\) 0 0
\(35\) −6.83172 + 1.09417i −1.15477 + 0.184949i
\(36\) 0 0
\(37\) 4.02893i 0.662352i 0.943569 + 0.331176i \(0.107445\pi\)
−0.943569 + 0.331176i \(0.892555\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 0.198578 0.0310127 0.0155064 0.999880i \(-0.495064\pi\)
0.0155064 + 0.999880i \(0.495064\pi\)
\(42\) 0 0
\(43\) 5.70861i 0.870555i 0.900296 + 0.435277i \(0.143350\pi\)
−0.900296 + 0.435277i \(0.856650\pi\)
\(44\) 0 0
\(45\) 0.353624 + 2.20793i 0.0527151 + 0.329139i
\(46\) 0 0
\(47\) 6.41586i 0.935849i −0.883768 0.467925i \(-0.845002\pi\)
0.883768 0.467925i \(-0.154998\pi\)
\(48\) 0 0
\(49\) −2.57391 −0.367702
\(50\) 0 0
\(51\) −6.21728 −0.870593
\(52\) 0 0
\(53\) 4.02893i 0.553416i 0.960954 + 0.276708i \(0.0892435\pi\)
−0.960954 + 0.276708i \(0.910756\pi\)
\(54\) 0 0
\(55\) 1.94848 + 12.1658i 0.262733 + 1.64043i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.53896 −0.981489 −0.490745 0.871303i \(-0.663275\pi\)
−0.490745 + 0.871303i \(0.663275\pi\)
\(60\) 0 0
\(61\) 11.9259 1.52695 0.763477 0.645835i \(-0.223491\pi\)
0.763477 + 0.645835i \(0.223491\pi\)
\(62\) 0 0
\(63\) 3.09417i 0.389829i
\(64\) 0 0
\(65\) 2.20793 0.353624i 0.273860 0.0438616i
\(66\) 0 0
\(67\) 4.83172i 0.590288i 0.955453 + 0.295144i \(0.0953676\pi\)
−0.955453 + 0.295144i \(0.904632\pi\)
\(68\) 0 0
\(69\) 4.21728 0.507701
\(70\) 0 0
\(71\) −7.98977 −0.948211 −0.474106 0.880468i \(-0.657228\pi\)
−0.474106 + 0.880468i \(0.657228\pi\)
\(72\) 0 0
\(73\) 9.31145i 1.08982i 0.838494 + 0.544912i \(0.183437\pi\)
−0.838494 + 0.544912i \(0.816563\pi\)
\(74\) 0 0
\(75\) −1.56155 4.74990i −0.180313 0.548471i
\(76\) 0 0
\(77\) 17.0490i 1.94291i
\(78\) 0 0
\(79\) 8.51139 0.957607 0.478803 0.877922i \(-0.341071\pi\)
0.478803 + 0.877922i \(0.341071\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.7926i 1.62369i 0.583871 + 0.811847i \(0.301538\pi\)
−0.583871 + 0.811847i \(0.698462\pi\)
\(84\) 0 0
\(85\) 13.7273 2.19858i 1.48894 0.238469i
\(86\) 0 0
\(87\) 1.70861i 0.183182i
\(88\) 0 0
\(89\) 9.40699 0.997139 0.498569 0.866850i \(-0.333859\pi\)
0.498569 + 0.866850i \(0.333859\pi\)
\(90\) 0 0
\(91\) 3.09417 0.324358
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.3026i 1.45221i −0.687585 0.726104i \(-0.741329\pi\)
0.687585 0.726104i \(-0.258671\pi\)
\(98\) 0 0
\(99\) 5.51003 0.553779
\(100\) 0 0
\(101\) −6.72595 −0.669257 −0.334628 0.942350i \(-0.608611\pi\)
−0.334628 + 0.942350i \(0.608611\pi\)
\(102\) 0 0
\(103\) 17.0779i 1.68274i 0.540461 + 0.841369i \(0.318250\pi\)
−0.540461 + 0.841369i \(0.681750\pi\)
\(104\) 0 0
\(105\) −1.09417 6.83172i −0.106780 0.666707i
\(106\) 0 0
\(107\) 7.63450i 0.738055i 0.929418 + 0.369027i \(0.120309\pi\)
−0.929418 + 0.369027i \(0.879691\pi\)
\(108\) 0 0
\(109\) 5.02006 0.480835 0.240417 0.970670i \(-0.422716\pi\)
0.240417 + 0.970670i \(0.422716\pi\)
\(110\) 0 0
\(111\) −4.02893 −0.382409
\(112\) 0 0
\(113\) 16.6229i 1.56375i 0.623434 + 0.781876i \(0.285737\pi\)
−0.623434 + 0.781876i \(0.714263\pi\)
\(114\) 0 0
\(115\) −9.31145 + 1.49133i −0.868297 + 0.139067i
\(116\) 0 0
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) 19.2373 1.76348
\(120\) 0 0
\(121\) 19.3604 1.76004
\(122\) 0 0
\(123\) 0.198578i 0.0179052i
\(124\) 0 0
\(125\) 5.12748 + 9.93524i 0.458615 + 0.888635i
\(126\) 0 0
\(127\) 2.87689i 0.255283i −0.991820 0.127642i \(-0.959259\pi\)
0.991820 0.127642i \(-0.0407407\pi\)
\(128\) 0 0
\(129\) −5.70861 −0.502615
\(130\) 0 0
\(131\) 16.5199 1.44335 0.721674 0.692233i \(-0.243373\pi\)
0.721674 + 0.692233i \(0.243373\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.20793 + 0.353624i −0.190028 + 0.0304351i
\(136\) 0 0
\(137\) 12.8504i 1.09789i −0.835860 0.548943i \(-0.815031\pi\)
0.835860 0.548943i \(-0.184969\pi\)
\(138\) 0 0
\(139\) −17.0490 −1.44608 −0.723038 0.690808i \(-0.757255\pi\)
−0.723038 + 0.690808i \(0.757255\pi\)
\(140\) 0 0
\(141\) 6.41586 0.540313
\(142\) 0 0
\(143\) 5.51003i 0.460772i
\(144\) 0 0
\(145\) −0.604205 3.77249i −0.0501765 0.313288i
\(146\) 0 0
\(147\) 2.57391i 0.212293i
\(148\) 0 0
\(149\) −19.5679 −1.60306 −0.801532 0.597952i \(-0.795981\pi\)
−0.801532 + 0.597952i \(0.795981\pi\)
\(150\) 0 0
\(151\) 9.22887 0.751035 0.375518 0.926815i \(-0.377465\pi\)
0.375518 + 0.926815i \(0.377465\pi\)
\(152\) 0 0
\(153\) 6.21728i 0.502637i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.89423i 0.310794i 0.987852 + 0.155397i \(0.0496656\pi\)
−0.987852 + 0.155397i \(0.950334\pi\)
\(158\) 0 0
\(159\) −4.02893 −0.319515
\(160\) 0 0
\(161\) −13.0490 −1.02840
\(162\) 0 0
\(163\) 0.237741i 0.0186213i 0.999957 + 0.00931067i \(0.00296372\pi\)
−0.999957 + 0.00931067i \(0.997036\pi\)
\(164\) 0 0
\(165\) −12.1658 + 1.94848i −0.947103 + 0.151689i
\(166\) 0 0
\(167\) 11.8304i 0.915460i −0.889091 0.457730i \(-0.848663\pi\)
0.889091 0.457730i \(-0.151337\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.6634i 0.886754i 0.896335 + 0.443377i \(0.146220\pi\)
−0.896335 + 0.443377i \(0.853780\pi\)
\(174\) 0 0
\(175\) 4.83172 + 14.6970i 0.365243 + 1.11099i
\(176\) 0 0
\(177\) 7.53896i 0.566663i
\(178\) 0 0
\(179\) −12.5403 −0.937308 −0.468654 0.883382i \(-0.655261\pi\)
−0.468654 + 0.883382i \(0.655261\pi\)
\(180\) 0 0
\(181\) −5.44343 −0.404607 −0.202303 0.979323i \(-0.564843\pi\)
−0.202303 + 0.979323i \(0.564843\pi\)
\(182\) 0 0
\(183\) 11.9259i 0.881587i
\(184\) 0 0
\(185\) 8.89560 1.42473i 0.654017 0.104748i
\(186\) 0 0
\(187\) 34.2574i 2.50515i
\(188\) 0 0
\(189\) −3.09417 −0.225068
\(190\) 0 0
\(191\) −5.22887 −0.378348 −0.189174 0.981944i \(-0.560581\pi\)
−0.189174 + 0.981944i \(0.560581\pi\)
\(192\) 0 0
\(193\) 17.5287i 1.26175i 0.775886 + 0.630873i \(0.217303\pi\)
−0.775886 + 0.630873i \(0.782697\pi\)
\(194\) 0 0
\(195\) 0.353624 + 2.20793i 0.0253235 + 0.158113i
\(196\) 0 0
\(197\) 22.4764i 1.60138i 0.599079 + 0.800690i \(0.295534\pi\)
−0.599079 + 0.800690i \(0.704466\pi\)
\(198\) 0 0
\(199\) −16.9143 −1.19902 −0.599511 0.800366i \(-0.704638\pi\)
−0.599511 + 0.800366i \(0.704638\pi\)
\(200\) 0 0
\(201\) −4.83172 −0.340803
\(202\) 0 0
\(203\) 5.28674i 0.371056i
\(204\) 0 0
\(205\) −0.0702221 0.438447i −0.00490452 0.0306225i
\(206\) 0 0
\(207\) 4.21728i 0.293121i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 11.8391 0.815037 0.407518 0.913197i \(-0.366394\pi\)
0.407518 + 0.913197i \(0.366394\pi\)
\(212\) 0 0
\(213\) 7.98977i 0.547450i
\(214\) 0 0
\(215\) 12.6042 2.01870i 0.859600 0.137674i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −9.31145 −0.629210
\(220\) 0 0
\(221\) −6.21728 −0.418219
\(222\) 0 0
\(223\) 4.48246i 0.300168i −0.988673 0.150084i \(-0.952046\pi\)
0.988673 0.150084i \(-0.0479544\pi\)
\(224\) 0 0
\(225\) 4.74990 1.56155i 0.316660 0.104104i
\(226\) 0 0
\(227\) 6.35799i 0.421995i −0.977487 0.210997i \(-0.932329\pi\)
0.977487 0.210997i \(-0.0676712\pi\)
\(228\) 0 0
\(229\) −6.45502 −0.426560 −0.213280 0.976991i \(-0.568415\pi\)
−0.213280 + 0.976991i \(0.568415\pi\)
\(230\) 0 0
\(231\) −17.0490 −1.12174
\(232\) 0 0
\(233\) 6.40290i 0.419468i −0.977758 0.209734i \(-0.932740\pi\)
0.977758 0.209734i \(-0.0672598\pi\)
\(234\) 0 0
\(235\) −14.1658 + 2.26880i −0.924072 + 0.148000i
\(236\) 0 0
\(237\) 8.51139i 0.552874i
\(238\) 0 0
\(239\) −1.55793 −0.100774 −0.0503872 0.998730i \(-0.516046\pi\)
−0.0503872 + 0.998730i \(0.516046\pi\)
\(240\) 0 0
\(241\) −14.1911 −0.914127 −0.457064 0.889434i \(-0.651099\pi\)
−0.457064 + 0.889434i \(0.651099\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0.910196 + 5.68301i 0.0581503 + 0.363074i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −14.7926 −0.937440
\(250\) 0 0
\(251\) −22.7082 −1.43333 −0.716665 0.697418i \(-0.754332\pi\)
−0.716665 + 0.697418i \(0.754332\pi\)
\(252\) 0 0
\(253\) 23.2373i 1.46092i
\(254\) 0 0
\(255\) 2.19858 + 13.7273i 0.137680 + 0.859638i
\(256\) 0 0
\(257\) 21.4172i 1.33597i 0.744175 + 0.667985i \(0.232843\pi\)
−0.744175 + 0.667985i \(0.767157\pi\)
\(258\) 0 0
\(259\) 12.4662 0.774613
\(260\) 0 0
\(261\) −1.70861 −0.105760
\(262\) 0 0
\(263\) 26.4951i 1.63376i 0.576807 + 0.816880i \(0.304298\pi\)
−0.576807 + 0.816880i \(0.695702\pi\)
\(264\) 0 0
\(265\) 8.89560 1.42473i 0.546452 0.0875203i
\(266\) 0 0
\(267\) 9.40699i 0.575698i
\(268\) 0 0
\(269\) −14.7259 −0.897857 −0.448928 0.893568i \(-0.648194\pi\)
−0.448928 + 0.893568i \(0.648194\pi\)
\(270\) 0 0
\(271\) 25.8491 1.57022 0.785109 0.619357i \(-0.212607\pi\)
0.785109 + 0.619357i \(0.212607\pi\)
\(272\) 0 0
\(273\) 3.09417i 0.187268i
\(274\) 0 0
\(275\) 26.1721 8.60421i 1.57824 0.518853i
\(276\) 0 0
\(277\) 20.1227i 1.20906i −0.796584 0.604528i \(-0.793362\pi\)
0.796584 0.604528i \(-0.206638\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −16.3954 −0.978067 −0.489034 0.872265i \(-0.662650\pi\)
−0.489034 + 0.872265i \(0.662650\pi\)
\(282\) 0 0
\(283\) 14.5650i 0.865802i 0.901441 + 0.432901i \(0.142510\pi\)
−0.901441 + 0.432901i \(0.857490\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.614436i 0.0362690i
\(288\) 0 0
\(289\) −21.6546 −1.27380
\(290\) 0 0
\(291\) 14.3026 0.838432
\(292\) 0 0
\(293\) 26.6995i 1.55980i −0.625904 0.779900i \(-0.715270\pi\)
0.625904 0.779900i \(-0.284730\pi\)
\(294\) 0 0
\(295\) 2.66596 + 16.6455i 0.155218 + 0.969138i
\(296\) 0 0
\(297\) 5.51003i 0.319724i
\(298\) 0 0
\(299\) 4.21728 0.243892
\(300\) 0 0
\(301\) 17.6634 1.01810
\(302\) 0 0
\(303\) 6.72595i 0.386396i
\(304\) 0 0
\(305\) −4.21728 26.3315i −0.241481 1.50774i
\(306\) 0 0
\(307\) 11.4230i 0.651943i 0.945380 + 0.325972i \(0.105691\pi\)
−0.945380 + 0.325972i \(0.894309\pi\)
\(308\) 0 0
\(309\) −17.0779 −0.971529
\(310\) 0 0
\(311\) −19.4546 −1.10317 −0.551585 0.834119i \(-0.685977\pi\)
−0.551585 + 0.834119i \(0.685977\pi\)
\(312\) 0 0
\(313\) 32.5805i 1.84156i −0.390087 0.920778i \(-0.627555\pi\)
0.390087 0.920778i \(-0.372445\pi\)
\(314\) 0 0
\(315\) 6.83172 1.09417i 0.384924 0.0616497i
\(316\) 0 0
\(317\) 31.3781i 1.76237i 0.472774 + 0.881184i \(0.343253\pi\)
−0.472774 + 0.881184i \(0.656747\pi\)
\(318\) 0 0
\(319\) −9.41450 −0.527111
\(320\) 0 0
\(321\) −7.63450 −0.426116
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.56155 4.74990i −0.0866194 0.263477i
\(326\) 0 0
\(327\) 5.02006i 0.277610i
\(328\) 0 0
\(329\) −19.8518 −1.09446
\(330\) 0 0
\(331\) 30.4951 1.67616 0.838082 0.545544i \(-0.183677\pi\)
0.838082 + 0.545544i \(0.183677\pi\)
\(332\) 0 0
\(333\) 4.02893i 0.220784i
\(334\) 0 0
\(335\) 10.6681 1.70861i 0.582860 0.0933513i
\(336\) 0 0
\(337\) 15.8765i 0.864848i 0.901670 + 0.432424i \(0.142342\pi\)
−0.901670 + 0.432424i \(0.857658\pi\)
\(338\) 0 0
\(339\) −16.6229 −0.902832
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.6951i 0.739465i
\(344\) 0 0
\(345\) −1.49133 9.31145i −0.0802905 0.501312i
\(346\) 0 0
\(347\) 21.1069i 1.13308i −0.824036 0.566538i \(-0.808283\pi\)
0.824036 0.566538i \(-0.191717\pi\)
\(348\) 0 0
\(349\) −4.82899 −0.258490 −0.129245 0.991613i \(-0.541255\pi\)
−0.129245 + 0.991613i \(0.541255\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 19.4359i 1.03447i −0.855844 0.517235i \(-0.826961\pi\)
0.855844 0.517235i \(-0.173039\pi\)
\(354\) 0 0
\(355\) 2.82537 + 17.6408i 0.149955 + 0.936279i
\(356\) 0 0
\(357\) 19.2373i 1.01815i
\(358\) 0 0
\(359\) −27.2271 −1.43699 −0.718496 0.695531i \(-0.755169\pi\)
−0.718496 + 0.695531i \(0.755169\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 19.3604i 1.01616i
\(364\) 0 0
\(365\) 20.5590 3.29275i 1.07611 0.172351i
\(366\) 0 0
\(367\) 15.7008i 0.819577i 0.912181 + 0.409788i \(0.134397\pi\)
−0.912181 + 0.409788i \(0.865603\pi\)
\(368\) 0 0
\(369\) −0.198578 −0.0103376
\(370\) 0 0
\(371\) 12.4662 0.647214
\(372\) 0 0
\(373\) 3.86952i 0.200356i −0.994970 0.100178i \(-0.968059\pi\)
0.994970 0.100178i \(-0.0319412\pi\)
\(374\) 0 0
\(375\) −9.93524 + 5.12748i −0.513054 + 0.264782i
\(376\) 0 0
\(377\) 1.70861i 0.0879979i
\(378\) 0 0
\(379\) −7.30149 −0.375052 −0.187526 0.982260i \(-0.560047\pi\)
−0.187526 + 0.982260i \(0.560047\pi\)
\(380\) 0 0
\(381\) 2.87689 0.147388
\(382\) 0 0
\(383\) 37.2094i 1.90131i −0.310250 0.950655i \(-0.600413\pi\)
0.310250 0.950655i \(-0.399587\pi\)
\(384\) 0 0
\(385\) 37.6430 6.02893i 1.91846 0.307263i
\(386\) 0 0
\(387\) 5.70861i 0.290185i
\(388\) 0 0
\(389\) −30.1227 −1.52728 −0.763641 0.645641i \(-0.776590\pi\)
−0.763641 + 0.645641i \(0.776590\pi\)
\(390\) 0 0
\(391\) 26.2200 1.32600
\(392\) 0 0
\(393\) 16.5199i 0.833317i
\(394\) 0 0
\(395\) −3.00983 18.7926i −0.151441 0.945556i
\(396\) 0 0
\(397\) 1.25780i 0.0631274i −0.999502 0.0315637i \(-0.989951\pi\)
0.999502 0.0315637i \(-0.0100487\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 20.6443 1.03093 0.515464 0.856911i \(-0.327619\pi\)
0.515464 + 0.856911i \(0.327619\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.353624 2.20793i −0.0175717 0.109713i
\(406\) 0 0
\(407\) 22.1995i 1.10039i
\(408\) 0 0
\(409\) 31.3820 1.55174 0.775870 0.630893i \(-0.217311\pi\)
0.775870 + 0.630893i \(0.217311\pi\)
\(410\) 0 0
\(411\) 12.8504 0.633864
\(412\) 0 0
\(413\) 23.3269i 1.14784i
\(414\) 0 0
\(415\) 32.6609 5.23100i 1.60326 0.256780i
\(416\) 0 0
\(417\) 17.0490i 0.834893i
\(418\) 0 0
\(419\) −19.5549 −0.955321 −0.477661 0.878544i \(-0.658515\pi\)
−0.477661 + 0.878544i \(0.658515\pi\)
\(420\) 0 0
\(421\) −5.07793 −0.247483 −0.123742 0.992314i \(-0.539489\pi\)
−0.123742 + 0.992314i \(0.539489\pi\)
\(422\) 0 0
\(423\) 6.41586i 0.311950i
\(424\) 0 0
\(425\) −9.70861 29.5315i −0.470937 1.43249i
\(426\) 0 0
\(427\) 36.9008i 1.78575i
\(428\) 0 0
\(429\) 5.51003 0.266027
\(430\) 0 0
\(431\) −7.52900 −0.362659 −0.181330 0.983422i \(-0.558040\pi\)
−0.181330 + 0.983422i \(0.558040\pi\)
\(432\) 0 0
\(433\) 37.4821i 1.80127i −0.434573 0.900637i \(-0.643101\pi\)
0.434573 0.900637i \(-0.356899\pi\)
\(434\) 0 0
\(435\) 3.77249 0.604205i 0.180877 0.0289694i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 13.4836 0.643535 0.321767 0.946819i \(-0.395723\pi\)
0.321767 + 0.946819i \(0.395723\pi\)
\(440\) 0 0
\(441\) 2.57391 0.122567
\(442\) 0 0
\(443\) 6.61171i 0.314132i 0.987588 + 0.157066i \(0.0502035\pi\)
−0.987588 + 0.157066i \(0.949796\pi\)
\(444\) 0 0
\(445\) −3.32654 20.7700i −0.157693 0.984591i
\(446\) 0 0
\(447\) 19.5679i 0.925530i
\(448\) 0 0
\(449\) −5.24183 −0.247377 −0.123689 0.992321i \(-0.539472\pi\)
−0.123689 + 0.992321i \(0.539472\pi\)
\(450\) 0 0
\(451\) −1.09417 −0.0515226
\(452\) 0 0
\(453\) 9.22887i 0.433610i
\(454\) 0 0
\(455\) −1.09417 6.83172i −0.0512957 0.320276i
\(456\) 0 0
\(457\) 8.69702i 0.406829i 0.979093 + 0.203415i \(0.0652039\pi\)
−0.979093 + 0.203415i \(0.934796\pi\)
\(458\) 0 0
\(459\) 6.21728 0.290198
\(460\) 0 0
\(461\) −3.11287 −0.144981 −0.0724905 0.997369i \(-0.523095\pi\)
−0.0724905 + 0.997369i \(0.523095\pi\)
\(462\) 0 0
\(463\) 33.8328i 1.57234i 0.618008 + 0.786172i \(0.287940\pi\)
−0.618008 + 0.786172i \(0.712060\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.0112i 1.29620i 0.761554 + 0.648102i \(0.224437\pi\)
−0.761554 + 0.648102i \(0.775563\pi\)
\(468\) 0 0
\(469\) 14.9502 0.690335
\(470\) 0 0
\(471\) −3.89423 −0.179437
\(472\) 0 0
\(473\) 31.4546i 1.44629i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4.02893i 0.184472i
\(478\) 0 0
\(479\) 0.424328 0.0193880 0.00969401 0.999953i \(-0.496914\pi\)
0.00969401 + 0.999953i \(0.496914\pi\)
\(480\) 0 0
\(481\) −4.02893 −0.183703
\(482\) 0 0
\(483\) 13.0490i 0.593750i
\(484\) 0 0
\(485\) −31.5791 + 5.05774i −1.43393 + 0.229660i
\(486\) 0 0
\(487\) 32.5720i 1.47598i −0.674813 0.737989i \(-0.735776\pi\)
0.674813 0.737989i \(-0.264224\pi\)
\(488\) 0 0
\(489\) −0.237741 −0.0107510
\(490\) 0 0
\(491\) 1.36659 0.0616734 0.0308367 0.999524i \(-0.490183\pi\)
0.0308367 + 0.999524i \(0.490183\pi\)
\(492\) 0 0
\(493\) 10.6229i 0.478432i
\(494\) 0 0
\(495\) −1.94848 12.1658i −0.0875776 0.546810i
\(496\) 0 0
\(497\) 24.7217i 1.10892i
\(498\) 0 0
\(499\) 17.6430 0.789808 0.394904 0.918722i \(-0.370778\pi\)
0.394904 + 0.918722i \(0.370778\pi\)
\(500\) 0 0
\(501\) 11.8304 0.528541
\(502\) 0 0
\(503\) 24.6202i 1.09776i −0.835901 0.548880i \(-0.815054\pi\)
0.835901 0.548880i \(-0.184946\pi\)
\(504\) 0 0
\(505\) 2.37846 + 14.8504i 0.105840 + 0.660835i
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 0 0
\(509\) −1.41204 −0.0625876 −0.0312938 0.999510i \(-0.509963\pi\)
−0.0312938 + 0.999510i \(0.509963\pi\)
\(510\) 0 0
\(511\) 28.8113 1.27453
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 37.7068 6.03916i 1.66156 0.266117i
\(516\) 0 0
\(517\) 35.3516i 1.55476i
\(518\) 0 0
\(519\) −11.6634 −0.511968
\(520\) 0 0
\(521\) 38.5298 1.68802 0.844011 0.536326i \(-0.180188\pi\)
0.844011 + 0.536326i \(0.180188\pi\)
\(522\) 0 0
\(523\) 2.23353i 0.0976653i 0.998807 + 0.0488326i \(0.0155501\pi\)
−0.998807 + 0.0488326i \(0.984450\pi\)
\(524\) 0 0
\(525\) −14.6970 + 4.83172i −0.641430 + 0.210873i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 5.21455 0.226720
\(530\) 0 0
\(531\) 7.53896 0.327163
\(532\) 0 0
\(533\) 0.198578i 0.00860139i
\(534\) 0 0
\(535\) 16.8564 2.69974i 0.728767 0.116720i
\(536\) 0 0
\(537\) 12.5403i 0.541155i
\(538\) 0 0
\(539\) 14.1823 0.610876
\(540\) 0 0
\(541\) −37.8892 −1.62898 −0.814492 0.580175i \(-0.802984\pi\)
−0.814492 + 0.580175i \(0.802984\pi\)
\(542\) 0 0
\(543\) 5.44343i 0.233600i
\(544\) 0 0
\(545\) −1.77521 11.0839i −0.0760418 0.474784i
\(546\) 0 0
\(547\) 43.8645i 1.87551i 0.347299 + 0.937755i \(0.387099\pi\)
−0.347299 + 0.937755i \(0.612901\pi\)
\(548\) 0 0
\(549\) −11.9259 −0.508985
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 26.3357i 1.11991i
\(554\) 0 0
\(555\) 1.42473 + 8.89560i 0.0604763 + 0.377597i
\(556\) 0 0
\(557\) 13.4537i 0.570050i 0.958520 + 0.285025i \(0.0920019\pi\)
−0.958520 + 0.285025i \(0.907998\pi\)
\(558\) 0 0
\(559\) −5.70861 −0.241448
\(560\) 0 0
\(561\) 34.2574 1.44635
\(562\) 0 0
\(563\) 31.6140i 1.33237i 0.745785 + 0.666186i \(0.232074\pi\)
−0.745785 + 0.666186i \(0.767926\pi\)
\(564\) 0 0
\(565\) 36.7022 5.87826i 1.54407 0.247300i
\(566\) 0 0
\(567\) 3.09417i 0.129943i
\(568\) 0 0
\(569\) 22.3562 0.937222 0.468611 0.883405i \(-0.344755\pi\)
0.468611 + 0.883405i \(0.344755\pi\)
\(570\) 0 0
\(571\) 12.6596 0.529788 0.264894 0.964277i \(-0.414663\pi\)
0.264894 + 0.964277i \(0.414663\pi\)
\(572\) 0 0
\(573\) 5.22887i 0.218439i
\(574\) 0 0
\(575\) 6.58550 + 20.0317i 0.274635 + 0.835378i
\(576\) 0 0
\(577\) 7.48861i 0.311755i 0.987776 + 0.155877i \(0.0498205\pi\)
−0.987776 + 0.155877i \(0.950180\pi\)
\(578\) 0 0
\(579\) −17.5287 −0.728469
\(580\) 0 0
\(581\) 45.7707 1.89889
\(582\) 0 0
\(583\) 22.1995i 0.919411i
\(584\) 0 0
\(585\) −2.20793 + 0.353624i −0.0912866 + 0.0146205i
\(586\) 0 0
\(587\) 4.41858i 0.182374i −0.995834 0.0911872i \(-0.970934\pi\)
0.995834 0.0911872i \(-0.0290662\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −22.4764 −0.924557
\(592\) 0 0
\(593\) 12.5810i 0.516641i 0.966059 + 0.258320i \(0.0831690\pi\)
−0.966059 + 0.258320i \(0.916831\pi\)
\(594\) 0 0
\(595\) −6.80278 42.4747i −0.278887 1.74129i
\(596\) 0 0
\(597\) 16.9143i 0.692256i
\(598\) 0 0
\(599\) −2.64337 −0.108005 −0.0540025 0.998541i \(-0.517198\pi\)
−0.0540025 + 0.998541i \(0.517198\pi\)
\(600\) 0 0
\(601\) −10.4740 −0.427243 −0.213622 0.976916i \(-0.568526\pi\)
−0.213622 + 0.976916i \(0.568526\pi\)
\(602\) 0 0
\(603\) 4.83172i 0.196763i
\(604\) 0 0
\(605\) −6.84632 42.7465i −0.278342 1.73789i
\(606\) 0 0
\(607\) 12.0197i 0.487863i 0.969792 + 0.243932i \(0.0784372\pi\)
−0.969792 + 0.243932i \(0.921563\pi\)
\(608\) 0 0
\(609\) 5.28674 0.214229
\(610\) 0 0
\(611\) 6.41586 0.259558
\(612\) 0 0
\(613\) 34.7676i 1.40425i 0.712054 + 0.702124i \(0.247765\pi\)
−0.712054 + 0.702124i \(0.752235\pi\)
\(614\) 0 0
\(615\) 0.438447 0.0702221i 0.0176799 0.00283163i
\(616\) 0 0
\(617\) 31.6644i 1.27476i −0.770549 0.637380i \(-0.780018\pi\)
0.770549 0.637380i \(-0.219982\pi\)
\(618\) 0 0
\(619\) 22.6024 0.908469 0.454234 0.890882i \(-0.349913\pi\)
0.454234 + 0.890882i \(0.349913\pi\)
\(620\) 0 0
\(621\) −4.21728 −0.169234
\(622\) 0 0
\(623\) 29.1069i 1.16614i
\(624\) 0 0
\(625\) 20.1231 14.8344i 0.804924 0.593378i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −25.0490 −0.998769
\(630\) 0 0
\(631\) −18.6229 −0.741366 −0.370683 0.928759i \(-0.620876\pi\)
−0.370683 + 0.928759i \(0.620876\pi\)
\(632\) 0 0
\(633\) 11.8391i 0.470562i
\(634\) 0 0
\(635\) −6.35198 + 1.01734i −0.252071 + 0.0403718i
\(636\) 0 0
\(637\) 2.57391i 0.101982i
\(638\) 0 0
\(639\) 7.98977 0.316070
\(640\) 0 0
\(641\) −8.77658 −0.346654 −0.173327 0.984864i \(-0.555452\pi\)
−0.173327 + 0.984864i \(0.555452\pi\)
\(642\) 0 0
\(643\) 30.6920i 1.21037i 0.796084 + 0.605186i \(0.206901\pi\)
−0.796084 + 0.605186i \(0.793099\pi\)
\(644\) 0 0
\(645\) 2.01870 + 12.6042i 0.0794863 + 0.496290i
\(646\) 0 0
\(647\) 36.2023i 1.42326i 0.702555 + 0.711629i \(0.252042\pi\)
−0.702555 + 0.711629i \(0.747958\pi\)
\(648\) 0 0
\(649\) 41.5399 1.63058
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.8746i 1.67781i −0.544277 0.838906i \(-0.683196\pi\)
0.544277 0.838906i \(-0.316804\pi\)
\(654\) 0 0
\(655\) −5.84182 36.4747i −0.228259 1.42518i
\(656\) 0 0
\(657\) 9.31145i 0.363274i
\(658\) 0 0
\(659\) −3.72907 −0.145264 −0.0726320 0.997359i \(-0.523140\pi\)
−0.0726320 + 0.997359i \(0.523140\pi\)
\(660\) 0 0
\(661\) −13.2057 −0.513642 −0.256821 0.966459i \(-0.582675\pi\)
−0.256821 + 0.966459i \(0.582675\pi\)
\(662\) 0 0
\(663\) 6.21728i 0.241459i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.20569i 0.279005i
\(668\) 0 0
\(669\) 4.48246 0.173302
\(670\) 0 0
\(671\) −65.7120 −2.53678
\(672\) 0 0
\(673\) 0.901611i 0.0347546i −0.999849 0.0173773i \(-0.994468\pi\)
0.999849 0.0173773i \(-0.00553164\pi\)
\(674\) 0 0
\(675\) 1.56155 + 4.74990i 0.0601042 + 0.182824i
\(676\) 0 0
\(677\) 10.0057i 0.384552i 0.981341 + 0.192276i \(0.0615869\pi\)
−0.981341 + 0.192276i \(0.938413\pi\)
\(678\) 0 0
\(679\) −44.2547 −1.69834
\(680\) 0 0
\(681\) 6.35799 0.243639
\(682\) 0 0
\(683\) 16.9110i 0.647082i 0.946214 + 0.323541i \(0.104873\pi\)
−0.946214 + 0.323541i \(0.895127\pi\)
\(684\) 0 0
\(685\) −28.3728 + 4.54421i −1.08407 + 0.173626i
\(686\) 0 0
\(687\) 6.45502i 0.246274i
\(688\) 0 0
\(689\) −4.02893 −0.153490
\(690\) 0 0
\(691\) −43.1003 −1.63961 −0.819807 0.572640i \(-0.805919\pi\)
−0.819807 + 0.572640i \(0.805919\pi\)
\(692\) 0 0
\(693\) 17.0490i 0.647638i
\(694\) 0 0
\(695\) 6.02893 + 37.6430i 0.228690 + 1.42788i
\(696\) 0 0
\(697\) 1.23462i 0.0467645i
\(698\) 0 0
\(699\) 6.40290 0.242180
\(700\) 0 0
\(701\) 34.9170 1.31880 0.659399 0.751793i \(-0.270811\pi\)
0.659399 + 0.751793i \(0.270811\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −2.26880 14.1658i −0.0854480 0.533513i
\(706\) 0 0
\(707\) 20.8113i 0.782688i
\(708\) 0 0
\(709\) −20.8140 −0.781685 −0.390843 0.920457i \(-0.627816\pi\)
−0.390843 + 0.920457i \(0.627816\pi\)
\(710\) 0 0
\(711\) −8.51139 −0.319202
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −12.1658 + 1.94848i −0.454974 + 0.0728690i
\(716\) 0 0
\(717\) 1.55793i 0.0581821i
\(718\) 0 0
\(719\) 10.6024 0.395404 0.197702 0.980262i \(-0.436652\pi\)
0.197702 + 0.980262i \(0.436652\pi\)
\(720\) 0 0
\(721\) 52.8421 1.96794
\(722\) 0 0
\(723\) 14.1911i 0.527772i
\(724\) 0 0
\(725\) −8.11573 + 2.66808i −0.301411 + 0.0990902i
\(726\) 0 0
\(727\) 1.06524i 0.0395076i −0.999805 0.0197538i \(-0.993712\pi\)
0.999805 0.0197538i \(-0.00628824\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −35.4920 −1.31272
\(732\) 0 0
\(733\) 29.1216i 1.07563i −0.843062 0.537816i \(-0.819250\pi\)
0.843062 0.537816i \(-0.180750\pi\)
\(734\) 0 0
\(735\) −5.68301 + 0.910196i −0.209621 + 0.0335731i
\(736\) 0 0
\(737\) 26.6229i 0.980667i
\(738\) 0 0
\(739\) −41.7966 −1.53751 −0.768757 0.639541i \(-0.779125\pi\)
−0.768757 + 0.639541i \(0.779125\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 32.8531i 1.20526i −0.798019 0.602632i \(-0.794119\pi\)
0.798019 0.602632i \(-0.205881\pi\)
\(744\) 0 0
\(745\) 6.91968 + 43.2045i 0.253517 + 1.58289i
\(746\) 0 0
\(747\) 14.7926i 0.541231i
\(748\) 0 0
\(749\) 23.6225 0.863146
\(750\) 0 0
\(751\) 20.2956 0.740597 0.370299 0.928913i \(-0.379255\pi\)
0.370299 + 0.928913i \(0.379255\pi\)
\(752\) 0 0
\(753\) 22.7082i 0.827533i
\(754\) 0 0
\(755\) −3.26355 20.3767i −0.118773 0.741584i
\(756\) 0 0
\(757\) 1.62331i 0.0590000i 0.999565 + 0.0295000i \(0.00939151\pi\)
−0.999565 + 0.0295000i \(0.990608\pi\)
\(758\) 0 0
\(759\) −23.2373 −0.843462
\(760\) 0 0
\(761\) −50.4410 −1.82848 −0.914242 0.405169i \(-0.867213\pi\)
−0.914242 + 0.405169i \(0.867213\pi\)
\(762\) 0 0
\(763\) 15.5329i 0.562330i
\(764\) 0 0
\(765\) −13.7273 + 2.19858i −0.496312 + 0.0794898i
\(766\) 0 0
\(767\) 7.53896i 0.272216i
\(768\) 0 0
\(769\) 6.75339 0.243533 0.121767 0.992559i \(-0.461144\pi\)
0.121767 + 0.992559i \(0.461144\pi\)
\(770\) 0 0
\(771\) −21.4172 −0.771322
\(772\) 0 0
\(773\) 36.5660i 1.31519i 0.753373 + 0.657594i \(0.228426\pi\)
−0.753373 + 0.657594i \(0.771574\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 12.4662i 0.447223i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 44.0239 1.57530
\(782\) 0 0
\(783\) 1.70861i 0.0610607i
\(784\) 0 0
\(785\) 8.59819 1.37709i 0.306883 0.0491506i
\(786\) 0 0
\(787\) 6.21426i 0.221514i 0.993847 + 0.110757i \(0.0353276\pi\)
−0.993847 + 0.110757i \(0.964672\pi\)
\(788\) 0 0
\(789\) −26.4951 −0.943252
\(790\) 0 0
\(791\) 51.4342 1.82879
\(792\) 0 0
\(793\) 11.9259i 0.423501i
\(794\) 0 0
\(795\) 1.42473 + 8.89560i 0.0505298 + 0.315494i
\(796\) 0 0
\(797\) 39.5788i 1.40195i −0.713184 0.700977i \(-0.752748\pi\)
0.713184 0.700977i \(-0.247252\pi\)
\(798\) 0 0
\(799\) 39.8892 1.41118
\(800\) 0 0
\(801\) −9.40699 −0.332380
\(802\) 0 0
\(803\) 51.3064i 1.81056i
\(804\)