Properties

Label 3150.2.b.f.251.6
Level 3150
Weight 2
Character 3150.251
Analytic conductor 25.153
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.7442857984.4
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.6
Root \(3.73923i\)
Character \(\chi\) = 3150.251
Dual form 3150.2.b.f.251.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +(-0.0951965 + 2.64404i) q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +(-0.0951965 + 2.64404i) q^{7} -1.00000i q^{8} -5.28808i q^{11} -2.19039i q^{13} +(-2.64404 - 0.0951965i) q^{14} +1.00000 q^{16} +1.04544 q^{17} +6.43303i q^{19} +5.28808 q^{22} +7.47847i q^{23} +2.19039 q^{26} +(0.0951965 - 2.64404i) q^{28} -7.47847i q^{29} -9.09768i q^{31} +1.00000i q^{32} +1.04544i q^{34} +0.855043 q^{37} -6.43303 q^{38} +2.19039 q^{41} +0.954564 q^{43} +5.28808i q^{44} -7.47847 q^{46} +11.0092 q^{47} +(-6.98188 - 0.503406i) q^{49} +2.19039i q^{52} -3.09768i q^{53} +(2.64404 + 0.0951965i) q^{56} +7.47847 q^{58} +13.7734 q^{59} +8.05225i q^{61} +9.09768 q^{62} -1.00000 q^{64} -5.33535 q^{67} -1.04544 q^{68} +6.43303i q^{71} -4.57615i q^{73} +0.855043i q^{74} -6.43303i q^{76} +(13.9819 + 0.503406i) q^{77} +15.6738 q^{79} +2.19039i q^{82} +4.38079 q^{83} +0.954564i q^{86} -5.28808 q^{88} +4.28126 q^{89} +(5.79148 + 0.208518i) q^{91} -7.47847i q^{92} +11.0092i q^{94} +11.8593i q^{97} +(0.503406 - 6.98188i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + 4q^{7} + O(q^{10}) \) \( 8q - 8q^{4} + 4q^{7} + 8q^{16} + 8q^{26} - 4q^{28} + 8q^{37} - 8q^{38} + 8q^{41} + 16q^{43} - 8q^{46} - 40q^{47} + 4q^{49} + 8q^{58} + 40q^{62} - 8q^{64} - 32q^{67} + 52q^{77} + 8q^{79} + 16q^{83} + 8q^{89} - 4q^{91} - 4q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −0.0951965 + 2.64404i −0.0359809 + 0.999352i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 5.28808i 1.59441i −0.603705 0.797207i \(-0.706310\pi\)
0.603705 0.797207i \(-0.293690\pi\)
\(12\) 0 0
\(13\) 2.19039i 0.607506i −0.952751 0.303753i \(-0.901760\pi\)
0.952751 0.303753i \(-0.0982397\pi\)
\(14\) −2.64404 0.0951965i −0.706649 0.0254423i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.04544 0.253555 0.126778 0.991931i \(-0.459537\pi\)
0.126778 + 0.991931i \(0.459537\pi\)
\(18\) 0 0
\(19\) 6.43303i 1.47584i 0.674889 + 0.737920i \(0.264192\pi\)
−0.674889 + 0.737920i \(0.735808\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.28808 1.12742
\(23\) 7.47847i 1.55937i 0.626173 + 0.779684i \(0.284620\pi\)
−0.626173 + 0.779684i \(0.715380\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.19039 0.429571
\(27\) 0 0
\(28\) 0.0951965 2.64404i 0.0179904 0.499676i
\(29\) 7.47847i 1.38872i −0.719629 0.694358i \(-0.755688\pi\)
0.719629 0.694358i \(-0.244312\pi\)
\(30\) 0 0
\(31\) 9.09768i 1.63399i −0.576643 0.816996i \(-0.695638\pi\)
0.576643 0.816996i \(-0.304362\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 1.04544i 0.179291i
\(35\) 0 0
\(36\) 0 0
\(37\) 0.855043 0.140568 0.0702841 0.997527i \(-0.477609\pi\)
0.0702841 + 0.997527i \(0.477609\pi\)
\(38\) −6.43303 −1.04358
\(39\) 0 0
\(40\) 0 0
\(41\) 2.19039 0.342082 0.171041 0.985264i \(-0.445287\pi\)
0.171041 + 0.985264i \(0.445287\pi\)
\(42\) 0 0
\(43\) 0.954564 0.145570 0.0727849 0.997348i \(-0.476811\pi\)
0.0727849 + 0.997348i \(0.476811\pi\)
\(44\) 5.28808i 0.797207i
\(45\) 0 0
\(46\) −7.47847 −1.10264
\(47\) 11.0092 1.60585 0.802927 0.596077i \(-0.203275\pi\)
0.802927 + 0.596077i \(0.203275\pi\)
\(48\) 0 0
\(49\) −6.98188 0.503406i −0.997411 0.0719152i
\(50\) 0 0
\(51\) 0 0
\(52\) 2.19039i 0.303753i
\(53\) 3.09768i 0.425500i −0.977107 0.212750i \(-0.931758\pi\)
0.977107 0.212750i \(-0.0682419\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.64404 + 0.0951965i 0.353324 + 0.0127212i
\(57\) 0 0
\(58\) 7.47847 0.981971
\(59\) 13.7734 1.79314 0.896569 0.442904i \(-0.146052\pi\)
0.896569 + 0.442904i \(0.146052\pi\)
\(60\) 0 0
\(61\) 8.05225i 1.03098i 0.856894 + 0.515492i \(0.172391\pi\)
−0.856894 + 0.515492i \(0.827609\pi\)
\(62\) 9.09768 1.15541
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.33535 −0.651817 −0.325908 0.945401i \(-0.605670\pi\)
−0.325908 + 0.945401i \(0.605670\pi\)
\(68\) −1.04544 −0.126778
\(69\) 0 0
\(70\) 0 0
\(71\) 6.43303i 0.763461i 0.924274 + 0.381730i \(0.124672\pi\)
−0.924274 + 0.381730i \(0.875328\pi\)
\(72\) 0 0
\(73\) 4.57615i 0.535598i −0.963475 0.267799i \(-0.913704\pi\)
0.963475 0.267799i \(-0.0862963\pi\)
\(74\) 0.855043i 0.0993967i
\(75\) 0 0
\(76\) 6.43303i 0.737920i
\(77\) 13.9819 + 0.503406i 1.59338 + 0.0573684i
\(78\) 0 0
\(79\) 15.6738 1.76344 0.881722 0.471769i \(-0.156384\pi\)
0.881722 + 0.471769i \(0.156384\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2.19039i 0.241888i
\(83\) 4.38079 0.480854 0.240427 0.970667i \(-0.422713\pi\)
0.240427 + 0.970667i \(0.422713\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.954564i 0.102933i
\(87\) 0 0
\(88\) −5.28808 −0.563711
\(89\) 4.28126 0.453813 0.226907 0.973917i \(-0.427139\pi\)
0.226907 + 0.973917i \(0.427139\pi\)
\(90\) 0 0
\(91\) 5.79148 + 0.208518i 0.607112 + 0.0218586i
\(92\) 7.47847i 0.779684i
\(93\) 0 0
\(94\) 11.0092i 1.13551i
\(95\) 0 0
\(96\) 0 0
\(97\) 11.8593i 1.20412i 0.798449 + 0.602062i \(0.205654\pi\)
−0.798449 + 0.602062i \(0.794346\pi\)
\(98\) 0.503406 6.98188i 0.0508517 0.705276i
\(99\) 0 0
\(100\) 0 0
\(101\) 11.5830 1.15255 0.576274 0.817257i \(-0.304506\pi\)
0.576274 + 0.817257i \(0.304506\pi\)
\(102\) 0 0
\(103\) 16.6757i 1.64310i −0.570134 0.821552i \(-0.693109\pi\)
0.570134 0.821552i \(-0.306891\pi\)
\(104\) −2.19039 −0.214786
\(105\) 0 0
\(106\) 3.09768 0.300874
\(107\) 7.00681i 0.677374i 0.940899 + 0.338687i \(0.109983\pi\)
−0.940899 + 0.338687i \(0.890017\pi\)
\(108\) 0 0
\(109\) −4.28991 −0.410899 −0.205450 0.978668i \(-0.565866\pi\)
−0.205450 + 0.978668i \(0.565866\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.0951965 + 2.64404i −0.00899522 + 0.249838i
\(113\) 8.09087i 0.761125i −0.924755 0.380563i \(-0.875730\pi\)
0.924755 0.380563i \(-0.124270\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 7.47847i 0.694358i
\(117\) 0 0
\(118\) 13.7734i 1.26794i
\(119\) −0.0995218 + 2.76417i −0.00912314 + 0.253391i
\(120\) 0 0
\(121\) −16.9638 −1.54216
\(122\) −8.05225 −0.729016
\(123\) 0 0
\(124\) 9.09768i 0.816996i
\(125\) 0 0
\(126\) 0 0
\(127\) −8.57118 −0.760569 −0.380285 0.924870i \(-0.624174\pi\)
−0.380285 + 0.924870i \(0.624174\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 5.90048 0.515527 0.257764 0.966208i \(-0.417014\pi\)
0.257764 + 0.966208i \(0.417014\pi\)
\(132\) 0 0
\(133\) −17.0092 0.612402i −1.47488 0.0531020i
\(134\) 5.33535i 0.460904i
\(135\) 0 0
\(136\) 1.04544i 0.0896454i
\(137\) 6.86607i 0.586608i −0.956019 0.293304i \(-0.905245\pi\)
0.956019 0.293304i \(-0.0947548\pi\)
\(138\) 0 0
\(139\) 13.9115i 1.17996i 0.807418 + 0.589979i \(0.200864\pi\)
−0.807418 + 0.589979i \(0.799136\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.43303 −0.539848
\(143\) −11.5830 −0.968616
\(144\) 0 0
\(145\) 0 0
\(146\) 4.57615 0.378725
\(147\) 0 0
\(148\) −0.855043 −0.0702841
\(149\) 15.9638i 1.30780i 0.756580 + 0.653901i \(0.226869\pi\)
−0.756580 + 0.653901i \(0.773131\pi\)
\(150\) 0 0
\(151\) 16.0546 1.30651 0.653253 0.757139i \(-0.273404\pi\)
0.653253 + 0.757139i \(0.273404\pi\)
\(152\) 6.43303 0.521788
\(153\) 0 0
\(154\) −0.503406 + 13.9819i −0.0405656 + 1.12669i
\(155\) 0 0
\(156\) 0 0
\(157\) 24.7665i 1.97659i 0.152569 + 0.988293i \(0.451245\pi\)
−0.152569 + 0.988293i \(0.548755\pi\)
\(158\) 15.6738i 1.24694i
\(159\) 0 0
\(160\) 0 0
\(161\) −19.7734 0.711924i −1.55836 0.0561075i
\(162\) 0 0
\(163\) −7.42622 −0.581667 −0.290833 0.956774i \(-0.593932\pi\)
−0.290833 + 0.956774i \(0.593932\pi\)
\(164\) −2.19039 −0.171041
\(165\) 0 0
\(166\) 4.38079i 0.340015i
\(167\) −0.573779 −0.0444003 −0.0222002 0.999754i \(-0.507067\pi\)
−0.0222002 + 0.999754i \(0.507067\pi\)
\(168\) 0 0
\(169\) 8.20218 0.630937
\(170\) 0 0
\(171\) 0 0
\(172\) −0.954564 −0.0727849
\(173\) 9.96375 0.757530 0.378765 0.925493i \(-0.376349\pi\)
0.378765 + 0.925493i \(0.376349\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.28808i 0.398604i
\(177\) 0 0
\(178\) 4.28126i 0.320894i
\(179\) 1.77336i 0.132547i −0.997801 0.0662735i \(-0.978889\pi\)
0.997801 0.0662735i \(-0.0211110\pi\)
\(180\) 0 0
\(181\) 11.0092i 0.818306i 0.912466 + 0.409153i \(0.134176\pi\)
−0.912466 + 0.409153i \(0.865824\pi\)
\(182\) −0.208518 + 5.79148i −0.0154564 + 0.429293i
\(183\) 0 0
\(184\) 7.47847 0.551320
\(185\) 0 0
\(186\) 0 0
\(187\) 5.52834i 0.404272i
\(188\) −11.0092 −0.802927
\(189\) 0 0
\(190\) 0 0
\(191\) 5.56697i 0.402812i 0.979508 + 0.201406i \(0.0645510\pi\)
−0.979508 + 0.201406i \(0.935449\pi\)
\(192\) 0 0
\(193\) 8.47166 0.609803 0.304902 0.952384i \(-0.401376\pi\)
0.304902 + 0.952384i \(0.401376\pi\)
\(194\) −11.8593 −0.851445
\(195\) 0 0
\(196\) 6.98188 + 0.503406i 0.498705 + 0.0359576i
\(197\) 12.8661i 0.916669i −0.888780 0.458335i \(-0.848446\pi\)
0.888780 0.458335i \(-0.151554\pi\)
\(198\) 0 0
\(199\) 17.5830i 1.24642i −0.782053 0.623212i \(-0.785828\pi\)
0.782053 0.623212i \(-0.214172\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 11.5830i 0.814975i
\(203\) 19.7734 + 0.711924i 1.38782 + 0.0499673i
\(204\) 0 0
\(205\) 0 0
\(206\) 16.6757 1.16185
\(207\) 0 0
\(208\) 2.19039i 0.151876i
\(209\) 34.0184 2.35310
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 3.09768i 0.212750i
\(213\) 0 0
\(214\) −7.00681 −0.478976
\(215\) 0 0
\(216\) 0 0
\(217\) 24.0546 + 0.866067i 1.63293 + 0.0587925i
\(218\) 4.28991i 0.290550i
\(219\) 0 0
\(220\) 0 0
\(221\) 2.28991i 0.154036i
\(222\) 0 0
\(223\) 10.8710i 0.727979i −0.931403 0.363989i \(-0.881414\pi\)
0.931403 0.363989i \(-0.118586\pi\)
\(224\) −2.64404 0.0951965i −0.176662 0.00636058i
\(225\) 0 0
\(226\) 8.09087 0.538197
\(227\) 14.0909 0.935244 0.467622 0.883929i \(-0.345111\pi\)
0.467622 + 0.883929i \(0.345111\pi\)
\(228\) 0 0
\(229\) 14.5239i 0.959767i 0.877332 + 0.479883i \(0.159321\pi\)
−0.877332 + 0.479883i \(0.840679\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −7.47847 −0.490986
\(233\) 20.6806i 1.35483i −0.735599 0.677417i \(-0.763099\pi\)
0.735599 0.677417i \(-0.236901\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −13.7734 −0.896569
\(237\) 0 0
\(238\) −2.76417 0.0995218i −0.179175 0.00645104i
\(239\) 5.56697i 0.360097i −0.983658 0.180049i \(-0.942375\pi\)
0.983658 0.180049i \(-0.0576255\pi\)
\(240\) 0 0
\(241\) 11.3839i 0.733303i 0.930358 + 0.366651i \(0.119496\pi\)
−0.930358 + 0.366651i \(0.880504\pi\)
\(242\) 16.9638i 1.09047i
\(243\) 0 0
\(244\) 8.05225i 0.515492i
\(245\) 0 0
\(246\) 0 0
\(247\) 14.0909 0.896581
\(248\) −9.09768 −0.577703
\(249\) 0 0
\(250\) 0 0
\(251\) 1.71874 0.108486 0.0542428 0.998528i \(-0.482725\pi\)
0.0542428 + 0.998528i \(0.482725\pi\)
\(252\) 0 0
\(253\) 39.5467 2.48628
\(254\) 8.57118i 0.537804i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.18779 −0.136471 −0.0682354 0.997669i \(-0.521737\pi\)
−0.0682354 + 0.997669i \(0.521737\pi\)
\(258\) 0 0
\(259\) −0.0813970 + 2.26077i −0.00505777 + 0.140477i
\(260\) 0 0
\(261\) 0 0
\(262\) 5.90048i 0.364533i
\(263\) 17.3876i 1.07217i 0.844166 + 0.536083i \(0.180096\pi\)
−0.844166 + 0.536083i \(0.819904\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0.612402 17.0092i 0.0375488 1.04290i
\(267\) 0 0
\(268\) 5.33535 0.325908
\(269\) 20.3445 1.24043 0.620214 0.784433i \(-0.287046\pi\)
0.620214 + 0.784433i \(0.287046\pi\)
\(270\) 0 0
\(271\) 1.47847i 0.0898106i 0.998991 + 0.0449053i \(0.0142986\pi\)
−0.998991 + 0.0449053i \(0.985701\pi\)
\(272\) 1.04544 0.0633888
\(273\) 0 0
\(274\) 6.86607 0.414794
\(275\) 0 0
\(276\) 0 0
\(277\) −10.7642 −0.646756 −0.323378 0.946270i \(-0.604819\pi\)
−0.323378 + 0.946270i \(0.604819\pi\)
\(278\) −13.9115 −0.834356
\(279\) 0 0
\(280\) 0 0
\(281\) 2.42806i 0.144846i −0.997374 0.0724230i \(-0.976927\pi\)
0.997374 0.0724230i \(-0.0230731\pi\)
\(282\) 0 0
\(283\) 24.5898i 1.46171i 0.682532 + 0.730855i \(0.260879\pi\)
−0.682532 + 0.730855i \(0.739121\pi\)
\(284\) 6.43303i 0.381730i
\(285\) 0 0
\(286\) 11.5830i 0.684915i
\(287\) −0.208518 + 5.79148i −0.0123084 + 0.341860i
\(288\) 0 0
\(289\) −15.9071 −0.935710
\(290\) 0 0
\(291\) 0 0
\(292\) 4.57615i 0.267799i
\(293\) −12.6707 −0.740230 −0.370115 0.928986i \(-0.620682\pi\)
−0.370115 + 0.928986i \(0.620682\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.855043i 0.0496983i
\(297\) 0 0
\(298\) −15.9638 −0.924755
\(299\) 16.3808 0.947325
\(300\) 0 0
\(301\) −0.0908711 + 2.52390i −0.00523773 + 0.145475i
\(302\) 16.0546i 0.923840i
\(303\) 0 0
\(304\) 6.43303i 0.368960i
\(305\) 0 0
\(306\) 0 0
\(307\) 0.866067i 0.0494291i −0.999695 0.0247145i \(-0.992132\pi\)
0.999695 0.0247145i \(-0.00786768\pi\)
\(308\) −13.9819 0.503406i −0.796691 0.0286842i
\(309\) 0 0
\(310\) 0 0
\(311\) −9.71009 −0.550608 −0.275304 0.961357i \(-0.588779\pi\)
−0.275304 + 0.961357i \(0.588779\pi\)
\(312\) 0 0
\(313\) 5.80096i 0.327889i −0.986470 0.163945i \(-0.947578\pi\)
0.986470 0.163945i \(-0.0524219\pi\)
\(314\) −24.7665 −1.39766
\(315\) 0 0
\(316\) −15.6738 −0.881722
\(317\) 28.1591i 1.58157i 0.612092 + 0.790787i \(0.290328\pi\)
−0.612092 + 0.790787i \(0.709672\pi\)
\(318\) 0 0
\(319\) −39.5467 −2.21419
\(320\) 0 0
\(321\) 0 0
\(322\) 0.711924 19.7734i 0.0396740 1.10193i
\(323\) 6.72532i 0.374207i
\(324\) 0 0
\(325\) 0 0
\(326\) 7.42622i 0.411300i
\(327\) 0 0
\(328\) 2.19039i 0.120944i
\(329\) −1.04804 + 29.1087i −0.0577801 + 1.60482i
\(330\) 0 0
\(331\) −19.5830 −1.07638 −0.538189 0.842824i \(-0.680891\pi\)
−0.538189 + 0.842824i \(0.680891\pi\)
\(332\) −4.38079 −0.240427
\(333\) 0 0
\(334\) 0.573779i 0.0313958i
\(335\) 0 0
\(336\) 0 0
\(337\) 28.3992 1.54700 0.773500 0.633796i \(-0.218504\pi\)
0.773500 + 0.633796i \(0.218504\pi\)
\(338\) 8.20218i 0.446140i
\(339\) 0 0
\(340\) 0 0
\(341\) −48.1092 −2.60526
\(342\) 0 0
\(343\) 1.99567 18.4124i 0.107756 0.994177i
\(344\) 0.954564i 0.0514667i
\(345\) 0 0
\(346\) 9.96375i 0.535655i
\(347\) 12.5898i 0.675855i −0.941172 0.337927i \(-0.890274\pi\)
0.941172 0.337927i \(-0.109726\pi\)
\(348\) 0 0
\(349\) 20.9956i 1.12387i −0.827183 0.561933i \(-0.810058\pi\)
0.827183 0.561933i \(-0.189942\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.28808 0.281855
\(353\) 15.1363 0.805624 0.402812 0.915283i \(-0.368033\pi\)
0.402812 + 0.915283i \(0.368033\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −4.28126 −0.226907
\(357\) 0 0
\(358\) 1.77336 0.0937249
\(359\) 33.1909i 1.75175i −0.482538 0.875875i \(-0.660285\pi\)
0.482538 0.875875i \(-0.339715\pi\)
\(360\) 0 0
\(361\) −22.3839 −1.17810
\(362\) −11.0092 −0.578630
\(363\) 0 0
\(364\) −5.79148 0.208518i −0.303556 0.0109293i
\(365\) 0 0
\(366\) 0 0
\(367\) 11.6825i 0.609821i 0.952381 + 0.304910i \(0.0986265\pi\)
−0.952381 + 0.304910i \(0.901373\pi\)
\(368\) 7.47847i 0.389842i
\(369\) 0 0
\(370\) 0 0
\(371\) 8.19039 + 0.294888i 0.425224 + 0.0153098i
\(372\) 0 0
\(373\) −12.8550 −0.665609 −0.332804 0.942996i \(-0.607995\pi\)
−0.332804 + 0.942996i \(0.607995\pi\)
\(374\) 5.52834 0.285864
\(375\) 0 0
\(376\) 11.0092i 0.567755i
\(377\) −16.3808 −0.843653
\(378\) 0 0
\(379\) 25.0751 1.28802 0.644010 0.765017i \(-0.277270\pi\)
0.644010 + 0.765017i \(0.277270\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −5.56697 −0.284831
\(383\) 23.0092 1.17571 0.587857 0.808965i \(-0.299972\pi\)
0.587857 + 0.808965i \(0.299972\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.47166i 0.431196i
\(387\) 0 0
\(388\) 11.8593i 0.602062i
\(389\) 10.1591i 0.515088i −0.966267 0.257544i \(-0.917087\pi\)
0.966267 0.257544i \(-0.0829132\pi\)
\(390\) 0 0
\(391\) 7.81826i 0.395386i
\(392\) −0.503406 + 6.98188i −0.0254258 + 0.352638i
\(393\) 0 0
\(394\) 12.8661 0.648183
\(395\) 0 0
\(396\) 0 0
\(397\) 13.9141i 0.698329i 0.937061 + 0.349164i \(0.113535\pi\)
−0.937061 + 0.349164i \(0.886465\pi\)
\(398\) 17.5830 0.881354
\(399\) 0 0
\(400\) 0 0
\(401\) 14.6197i 0.730075i 0.930993 + 0.365038i \(0.118944\pi\)
−0.930993 + 0.365038i \(0.881056\pi\)
\(402\) 0 0
\(403\) −19.9275 −0.992660
\(404\) −11.5830 −0.576274
\(405\) 0 0
\(406\) −0.711924 + 19.7734i −0.0353322 + 0.981335i
\(407\) 4.52153i 0.224124i
\(408\) 0 0
\(409\) 12.0000i 0.593362i −0.954977 0.296681i \(-0.904120\pi\)
0.954977 0.296681i \(-0.0958798\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 16.6757i 0.821552i
\(413\) −1.31117 + 36.4173i −0.0645187 + 1.79198i
\(414\) 0 0
\(415\) 0 0
\(416\) 2.19039 0.107393
\(417\) 0 0
\(418\) 34.0184i 1.66389i
\(419\) 2.60743 0.127381 0.0636906 0.997970i \(-0.479713\pi\)
0.0636906 + 0.997970i \(0.479713\pi\)
\(420\) 0 0
\(421\) 0.852443 0.0415455 0.0207728 0.999784i \(-0.493387\pi\)
0.0207728 + 0.999784i \(0.493387\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 0 0
\(424\) −3.09768 −0.150437
\(425\) 0 0
\(426\) 0 0
\(427\) −21.2905 0.766545i −1.03032 0.0370957i
\(428\) 7.00681i 0.338687i
\(429\) 0 0
\(430\) 0 0
\(431\) 20.2476i 0.975293i −0.873041 0.487647i \(-0.837855\pi\)
0.873041 0.487647i \(-0.162145\pi\)
\(432\) 0 0
\(433\) 30.6444i 1.47268i −0.676614 0.736338i \(-0.736553\pi\)
0.676614 0.736338i \(-0.263447\pi\)
\(434\) −0.866067 + 24.0546i −0.0415726 + 1.15466i
\(435\) 0 0
\(436\) 4.28991 0.205450
\(437\) −48.1092 −2.30138
\(438\) 0 0
\(439\) 22.6308i 1.08011i −0.841630 0.540054i \(-0.818404\pi\)
0.841630 0.540054i \(-0.181596\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.28991 0.108920
\(443\) 36.3083i 1.72506i −0.506007 0.862529i \(-0.668879\pi\)
0.506007 0.862529i \(-0.331121\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 10.8710 0.514759
\(447\) 0 0
\(448\) 0.0951965 2.64404i 0.00449761 0.124919i
\(449\) 25.6764i 1.21175i −0.795561 0.605873i \(-0.792824\pi\)
0.795561 0.605873i \(-0.207176\pi\)
\(450\) 0 0
\(451\) 11.5830i 0.545421i
\(452\) 8.09087i 0.380563i
\(453\) 0 0
\(454\) 14.0909i 0.661317i
\(455\) 0 0
\(456\) 0 0
\(457\) −25.3477 −1.18571 −0.592857 0.805308i \(-0.702000\pi\)
−0.592857 + 0.805308i \(0.702000\pi\)
\(458\) −14.5239 −0.678657
\(459\) 0 0
\(460\) 0 0
\(461\) −12.7253 −0.592677 −0.296339 0.955083i \(-0.595766\pi\)
−0.296339 + 0.955083i \(0.595766\pi\)
\(462\) 0 0
\(463\) 21.2655 0.988289 0.494145 0.869380i \(-0.335481\pi\)
0.494145 + 0.869380i \(0.335481\pi\)
\(464\) 7.47847i 0.347179i
\(465\) 0 0
\(466\) 20.6806 0.958013
\(467\) 13.1424 0.608156 0.304078 0.952647i \(-0.401652\pi\)
0.304078 + 0.952647i \(0.401652\pi\)
\(468\) 0 0
\(469\) 0.507906 14.1069i 0.0234529 0.651395i
\(470\) 0 0
\(471\) 0 0
\(472\) 13.7734i 0.633970i
\(473\) 5.04781i 0.232099i
\(474\) 0 0
\(475\) 0 0
\(476\) 0.0995218 2.76417i 0.00456157 0.126696i
\(477\) 0 0
\(478\) 5.56697 0.254627
\(479\) −26.2899 −1.20122 −0.600608 0.799543i \(-0.705075\pi\)
−0.600608 + 0.799543i \(0.705075\pi\)
\(480\) 0 0
\(481\) 1.87288i 0.0853959i
\(482\) −11.3839 −0.518523
\(483\) 0 0
\(484\) 16.9638 0.771080
\(485\) 0 0
\(486\) 0 0
\(487\) −39.5381 −1.79164 −0.895820 0.444416i \(-0.853411\pi\)
−0.895820 + 0.444416i \(0.853411\pi\)
\(488\) 8.05225 0.364508
\(489\) 0 0
\(490\) 0 0
\(491\) 4.62105i 0.208545i 0.994549 + 0.104273i \(0.0332514\pi\)
−0.994549 + 0.104273i \(0.966749\pi\)
\(492\) 0 0
\(493\) 7.81826i 0.352117i
\(494\) 14.0909i 0.633978i
\(495\) 0 0
\(496\) 9.09768i 0.408498i
\(497\) −17.0092 0.612402i −0.762966 0.0274700i
\(498\) 0 0
\(499\) 31.0152 1.38843 0.694216 0.719766i \(-0.255751\pi\)
0.694216 + 0.719766i \(0.255751\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1.71874i 0.0767109i
\(503\) −28.3385 −1.26355 −0.631775 0.775152i \(-0.717673\pi\)
−0.631775 + 0.775152i \(0.717673\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 39.5467i 1.75807i
\(507\) 0 0
\(508\) 8.57118 0.380285
\(509\) −22.3808 −0.992011 −0.496005 0.868319i \(-0.665200\pi\)
−0.496005 + 0.868319i \(0.665200\pi\)
\(510\) 0 0
\(511\) 12.0995 + 0.435633i 0.535251 + 0.0192713i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 2.18779i 0.0964994i
\(515\) 0 0
\(516\) 0 0
\(517\) 58.2174i 2.56040i
\(518\) −2.26077 0.0813970i −0.0993323 0.00357638i
\(519\) 0 0
\(520\) 0 0
\(521\) −28.5896 −1.25253 −0.626265 0.779610i \(-0.715417\pi\)
−0.626265 + 0.779610i \(0.715417\pi\)
\(522\) 0 0
\(523\) 3.51472i 0.153688i 0.997043 + 0.0768440i \(0.0244843\pi\)
−0.997043 + 0.0768440i \(0.975516\pi\)
\(524\) −5.90048 −0.257764
\(525\) 0 0
\(526\) −17.3876 −0.758135
\(527\) 9.51104i 0.414307i
\(528\) 0 0
\(529\) −32.9275 −1.43163
\(530\) 0 0
\(531\) 0 0
\(532\) 17.0092 + 0.612402i 0.737442 + 0.0265510i
\(533\) 4.79782i 0.207817i
\(534\) 0 0
\(535\) 0 0
\(536\) 5.33535i 0.230452i
\(537\) 0 0
\(538\) 20.3445i 0.877115i
\(539\) −2.66205 + 36.9207i −0.114663 + 1.59029i
\(540\) 0 0
\(541\) 30.4900 1.31087 0.655434 0.755252i \(-0.272486\pi\)
0.655434 + 0.755252i \(0.272486\pi\)
\(542\) −1.47847 −0.0635057
\(543\) 0 0
\(544\) 1.04544i 0.0448227i
\(545\) 0 0
\(546\) 0 0
\(547\) 6.86369 0.293470 0.146735 0.989176i \(-0.453123\pi\)
0.146735 + 0.989176i \(0.453123\pi\)
\(548\) 6.86607i 0.293304i
\(549\) 0 0
\(550\) 0 0
\(551\) 48.1092 2.04952
\(552\) 0 0
\(553\) −1.49209 + 41.4422i −0.0634503 + 1.76230i
\(554\) 10.7642i 0.457326i
\(555\) 0 0
\(556\) 13.9115i 0.589979i
\(557\) 1.81826i 0.0770421i −0.999258 0.0385210i \(-0.987735\pi\)
0.999258 0.0385210i \(-0.0122647\pi\)
\(558\) 0 0
\(559\) 2.09087i 0.0884344i
\(560\) 0 0
\(561\) 0 0
\(562\) 2.42806 0.102422
\(563\) 34.0184 1.43370 0.716852 0.697226i \(-0.245582\pi\)
0.716852 + 0.697226i \(0.245582\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −24.5898 −1.03359
\(567\) 0 0
\(568\) 6.43303 0.269924
\(569\) 5.10871i 0.214168i 0.994250 + 0.107084i \(0.0341514\pi\)
−0.994250 + 0.107084i \(0.965849\pi\)
\(570\) 0 0
\(571\) 10.8214 0.452861 0.226431 0.974027i \(-0.427294\pi\)
0.226431 + 0.974027i \(0.427294\pi\)
\(572\) 11.5830 0.484308
\(573\) 0 0
\(574\) −5.79148 0.208518i −0.241732 0.00870336i
\(575\) 0 0
\(576\) 0 0
\(577\) 39.1523i 1.62993i −0.579509 0.814966i \(-0.696756\pi\)
0.579509 0.814966i \(-0.303244\pi\)
\(578\) 15.9071i 0.661647i
\(579\) 0 0
\(580\) 0 0
\(581\) −0.417035 + 11.5830i −0.0173015 + 0.480542i
\(582\) 0 0
\(583\) −16.3808 −0.678423
\(584\) −4.57615 −0.189363
\(585\) 0 0
\(586\) 12.6707i 0.523422i
\(587\) 5.52834 0.228179 0.114090 0.993470i \(-0.463605\pi\)
0.114090 + 0.993470i \(0.463605\pi\)
\(588\) 0 0
\(589\) 58.5257 2.41151
\(590\) 0 0
\(591\) 0 0
\(592\) 0.855043 0.0351420
\(593\) −18.6830 −0.767220 −0.383610 0.923495i \(-0.625319\pi\)
−0.383610 + 0.923495i \(0.625319\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.9638i 0.653901i
\(597\) 0 0
\(598\) 16.3808i 0.669860i
\(599\) 4.06587i 0.166127i −0.996544 0.0830635i \(-0.973530\pi\)
0.996544 0.0830635i \(-0.0264704\pi\)
\(600\) 0 0
\(601\) 25.1161i 1.02451i 0.858835 + 0.512253i \(0.171189\pi\)
−0.858835 + 0.512253i \(0.828811\pi\)
\(602\) −2.52390 0.0908711i −0.102867 0.00370363i
\(603\) 0 0
\(604\) −16.0546 −0.653253
\(605\) 0 0
\(606\) 0 0
\(607\) 31.4336i 1.27585i −0.770099 0.637925i \(-0.779793\pi\)
0.770099 0.637925i \(-0.220207\pi\)
\(608\) −6.43303 −0.260894
\(609\) 0 0
\(610\) 0 0
\(611\) 24.1144i 0.975566i
\(612\) 0 0
\(613\) −39.4532 −1.59350 −0.796751 0.604308i \(-0.793450\pi\)
−0.796751 + 0.604308i \(0.793450\pi\)
\(614\) 0.866067 0.0349516
\(615\) 0 0
\(616\) 0.503406 13.9819i 0.0202828 0.563346i
\(617\) 39.7421i 1.59996i −0.600029 0.799978i \(-0.704844\pi\)
0.600029 0.799978i \(-0.295156\pi\)
\(618\) 0 0
\(619\) 14.7776i 0.593961i −0.954884 0.296980i \(-0.904020\pi\)
0.954884 0.296980i \(-0.0959796\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 9.71009i 0.389339i
\(623\) −0.407561 + 11.3198i −0.0163286 + 0.453519i
\(624\) 0 0
\(625\) 0 0
\(626\) 5.80096 0.231853
\(627\) 0 0
\(628\) 24.7665i 0.988293i
\(629\) 0.893892 0.0356418
\(630\) 0 0
\(631\) −16.3083 −0.649223 −0.324611 0.945847i \(-0.605233\pi\)
−0.324611 + 0.945847i \(0.605233\pi\)
\(632\) 15.6738i 0.623472i
\(633\) 0 0
\(634\) −28.1591 −1.11834
\(635\) 0 0
\(636\) 0 0
\(637\) −1.10266 + 15.2931i −0.0436889 + 0.605933i
\(638\) 39.5467i 1.56567i
\(639\) 0 0
\(640\) 0 0
\(641\) 12.5289i 0.494861i 0.968906 + 0.247430i \(0.0795862\pi\)
−0.968906 + 0.247430i \(0.920414\pi\)
\(642\) 0 0
\(643\) 14.3992i 0.567847i 0.958847 + 0.283924i \(0.0916362\pi\)
−0.958847 + 0.283924i \(0.908364\pi\)
\(644\) 19.7734 + 0.711924i 0.779179 + 0.0280537i
\(645\) 0 0
\(646\) −6.72532 −0.264604
\(647\) 31.7708 1.24904 0.624519 0.781010i \(-0.285295\pi\)
0.624519 + 0.781010i \(0.285295\pi\)
\(648\) 0 0
\(649\) 72.8346i 2.85901i
\(650\) 0 0
\(651\) 0 0
\(652\) 7.42622 0.290833
\(653\) 20.7389i 0.811578i −0.913967 0.405789i \(-0.866997\pi\)
0.913967 0.405789i \(-0.133003\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.19039 0.0855205
\(657\) 0 0
\(658\) −29.1087 1.04804i −1.13478 0.0408567i
\(659\) 2.44038i 0.0950638i 0.998870 + 0.0475319i \(0.0151356\pi\)
−0.998870 + 0.0475319i \(0.984864\pi\)
\(660\) 0 0
\(661\) 33.3863i 1.29858i −0.760542 0.649288i \(-0.775067\pi\)
0.760542 0.649288i \(-0.224933\pi\)
\(662\) 19.5830i 0.761114i
\(663\) 0 0
\(664\) 4.38079i 0.170007i
\(665\) 0 0
\(666\) 0 0
\(667\) 55.9275 2.16552
\(668\) 0.573779 0.0222002
\(669\) 0 0
\(670\) 0 0
\(671\) 42.5809 1.64382
\(672\) 0 0
\(673\) 18.6943 0.720611 0.360306 0.932834i \(-0.382672\pi\)
0.360306 + 0.932834i \(0.382672\pi\)
\(674\) 28.3992i 1.09389i
\(675\) 0 0
\(676\) −8.20218 −0.315468
\(677\) −36.5861 −1.40612 −0.703059 0.711131i \(-0.748183\pi\)
−0.703059 + 0.711131i \(0.748183\pi\)
\(678\) 0 0
\(679\) −31.3563 1.12896i −1.20335 0.0433255i
\(680\) 0 0
\(681\) 0 0
\(682\) 48.1092i 1.84220i
\(683\) 36.9207i 1.41273i 0.707847 + 0.706365i \(0.249666\pi\)
−0.707847 + 0.706365i \(0.750334\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 18.4124 + 1.99567i 0.702990 + 0.0761952i
\(687\) 0 0
\(688\) 0.954564 0.0363924
\(689\) −6.78514 −0.258493
\(690\) 0 0
\(691\) 5.20823i 0.198130i 0.995081 + 0.0990652i \(0.0315852\pi\)
−0.995081 + 0.0990652i \(0.968415\pi\)
\(692\) −9.96375 −0.378765
\(693\) 0 0
\(694\) 12.5898 0.477901
\(695\) 0 0
\(696\) 0 0
\(697\) 2.28991 0.0867367
\(698\) 20.9956 0.794694
\(699\) 0 0
\(700\) 0 0
\(701\) 13.2831i 0.501696i 0.968027 + 0.250848i \(0.0807094\pi\)
−0.968027 + 0.250848i \(0.919291\pi\)
\(702\) 0 0
\(703\) 5.50052i 0.207456i
\(704\) 5.28808i 0.199302i
\(705\) 0 0
\(706\) 15.1363i 0.569662i
\(707\) −1.10266 + 30.6258i −0.0414697 + 1.15180i
\(708\) 0 0
\(709\) −4.09087 −0.153636 −0.0768179 0.997045i \(-0.524476\pi\)
−0.0768179 + 0.997045i \(0.524476\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 4.28126i 0.160447i
\(713\) 68.0367 2.54800
\(714\) 0 0
\(715\) 0 0
\(716\) 1.77336i 0.0662735i
\(717\) 0 0
\(718\) 33.1909 1.23867
\(719\) −16.1817 −0.603477 −0.301739 0.953391i \(-0.597567\pi\)
−0.301739 + 0.953391i \(0.597567\pi\)
\(720\) 0 0
\(721\) 44.0911 + 1.58747i 1.64204 + 0.0591203i
\(722\) 22.3839i 0.833043i
\(723\) 0 0
\(724\) 11.0092i 0.409153i
\(725\) 0 0
\(726\) 0 0
\(727\) 40.7849i 1.51263i −0.654208 0.756314i \(-0.726998\pi\)
0.654208 0.756314i \(-0.273002\pi\)
\(728\) 0.208518 5.79148i 0.00772818 0.214647i
\(729\) 0 0
\(730\) 0 0
\(731\) 0.997936 0.0369100
\(732\) 0 0
\(733\) 39.2481i 1.44966i 0.688926 + 0.724832i \(0.258083\pi\)
−0.688926 + 0.724832i \(0.741917\pi\)
\(734\) −11.6825 −0.431208
\(735\) 0 0
\(736\) −7.47847 −0.275660
\(737\) 28.2137i 1.03927i
\(738\) 0 0
\(739\) −16.7305 −0.615442 −0.307721 0.951477i \(-0.599566\pi\)
−0.307721 + 0.951477i \(0.599566\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.294888 + 8.19039i −0.0108257 + 0.300679i
\(743\) 35.7185i 1.31039i −0.755462 0.655193i \(-0.772588\pi\)
0.755462 0.655193i \(-0.227412\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 12.8550i 0.470657i
\(747\) 0 0
\(748\) 5.52834i 0.202136i
\(749\) −18.5263 0.667024i −0.676935 0.0243725i
\(750\) 0 0
\(751\) −9.14236 −0.333609 −0.166805 0.985990i \(-0.553345\pi\)
−0.166805 + 0.985990i \(0.553345\pi\)
\(752\) 11.0092 0.401464
\(753\) 0 0
\(754\) 16.3808i 0.596553i
\(755\) 0 0
\(756\) 0 0
\(757\) −48.2027 −1.75196 −0.875979 0.482350i \(-0.839783\pi\)
−0.875979 + 0.482350i \(0.839783\pi\)
\(758\) 25.0751i 0.910767i
\(759\) 0 0
\(760\) 0 0
\(761\) −9.61056 −0.348383 −0.174191 0.984712i \(-0.555731\pi\)
−0.174191 + 0.984712i \(0.555731\pi\)
\(762\) 0 0
\(763\) 0.408385 11.3427i 0.0147845 0.410633i
\(764\) 5.56697i 0.201406i
\(765\) 0 0
\(766\) 23.0092i 0.831356i
\(767\) 30.1691i 1.08934i
\(768\) 0 0
\(769\) 33.2931i 1.20058i 0.799783 + 0.600289i \(0.204948\pi\)
−0.799783 + 0.600289i \(0.795052\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −8.47166 −0.304902
\(773\) −12.2726 −0.441415 −0.220708 0.975340i \(-0.570837\pi\)
−0.220708 + 0.975340i \(0.570837\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 11.8593 0.425722
\(777\) 0 0
\(778\) 10.1591 0.364222
\(779\) 14.0909i 0.504858i
\(780\) 0 0
\(781\) 34.0184 1.21727
\(782\) −7.81826 −0.279580
\(783\) 0 0
\(784\) −6.98188 0.503406i −0.249353 0.0179788i
\(785\) 0 0
\(786\) 0 0
\(787\) 21.4286i 0.763847i −0.924194 0.381923i \(-0.875262\pi\)
0.924194 0.381923i \(-0.124738\pi\)
\(788\) 12.8661i 0.458335i
\(789\) 0 0
\(790\) 0 0
\(791\) 21.3926 + 0.770222i 0.760632 + 0.0273859i
\(792\) 0 0
\(793\) 17.6376 0.626329
\(794\) −13.9141 −0.493793
\(795\) 0 0
\(796\) 17.5830i 0.623212i
\(797\) −9.96375 −0.352934 −0.176467 0.984307i \(-0.556467\pi\)
−0.176467 + 0.984307i \(0.556467\pi\)
\(798\) 0 0
\(799\) 11.5094 0.407173
\(800\) 0 0
\(801\) 0 0
\(802\) −14.6197 −0.516241
\(803\) −24.1990 −0.853966