Properties

Label 3150.2.d.c.3149.8
Level 3150
Weight 2
Character 3150.3149
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.d (of order \(2\), degree \(1\), not 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: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3149.8
Root \(-3.73923i\)
Character \(\chi\) = 3150.3149
Dual form 3150.2.d.c.3149.7

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +(2.64404 + 0.0951965i) q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +(2.64404 + 0.0951965i) q^{7} -1.00000 q^{8} +5.28808i q^{11} +2.19039 q^{13} +(-2.64404 - 0.0951965i) q^{14} +1.00000 q^{16} +1.04544i q^{17} -6.43303i q^{19} -5.28808i q^{22} +7.47847 q^{23} -2.19039 q^{26} +(2.64404 + 0.0951965i) q^{28} -7.47847i q^{29} -9.09768i q^{31} -1.00000 q^{32} -1.04544i q^{34} -0.855043i q^{37} +6.43303i q^{38} -2.19039 q^{41} +0.954564i q^{43} +5.28808i q^{44} -7.47847 q^{46} +11.0092i q^{47} +(6.98188 + 0.503406i) q^{49} +2.19039 q^{52} -3.09768 q^{53} +(-2.64404 - 0.0951965i) q^{56} +7.47847i q^{58} +13.7734 q^{59} +8.05225i q^{61} +9.09768i q^{62} +1.00000 q^{64} +5.33535i q^{67} +1.04544i q^{68} -6.43303i q^{71} +4.57615 q^{73} +0.855043i q^{74} -6.43303i q^{76} +(-0.503406 + 13.9819i) q^{77} -15.6738 q^{79} +2.19039 q^{82} -4.38079i q^{83} -0.954564i q^{86} -5.28808i q^{88} +4.28126 q^{89} +(5.79148 + 0.208518i) q^{91} +7.47847 q^{92} -11.0092i q^{94} +11.8593 q^{97} +(-6.98188 - 0.503406i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{2} + 8q^{4} - 8q^{8} + O(q^{10}) \) \( 8q - 8q^{2} + 8q^{4} - 8q^{8} + 8q^{13} + 8q^{16} + 8q^{23} - 8q^{26} - 8q^{32} - 8q^{41} - 8q^{46} - 4q^{49} + 8q^{52} + 8q^{53} + 8q^{64} - 48q^{73} + 4q^{77} - 8q^{79} + 8q^{82} + 8q^{89} - 4q^{91} + 8q^{92} + 24q^{97} + 4q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.64404 + 0.0951965i 0.999352 + 0.0359809i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 5.28808i 1.59441i 0.603705 + 0.797207i \(0.293690\pi\)
−0.603705 + 0.797207i \(0.706310\pi\)
\(12\) 0 0
\(13\) 2.19039 0.607506 0.303753 0.952751i \(-0.401760\pi\)
0.303753 + 0.952751i \(0.401760\pi\)
\(14\) −2.64404 0.0951965i −0.706649 0.0254423i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.04544i 0.253555i 0.991931 + 0.126778i \(0.0404635\pi\)
−0.991931 + 0.126778i \(0.959537\pi\)
\(18\) 0 0
\(19\) 6.43303i 1.47584i −0.674889 0.737920i \(-0.735808\pi\)
0.674889 0.737920i \(-0.264192\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.28808i 1.12742i
\(23\) 7.47847 1.55937 0.779684 0.626173i \(-0.215380\pi\)
0.779684 + 0.626173i \(0.215380\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.19039 −0.429571
\(27\) 0 0
\(28\) 2.64404 + 0.0951965i 0.499676 + 0.0179904i
\(29\) 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 \(-0.304362\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.04544i 0.179291i
\(35\) 0 0
\(36\) 0 0
\(37\) 0.855043i 0.140568i −0.997527 0.0702841i \(-0.977609\pi\)
0.997527 0.0702841i \(-0.0223906\pi\)
\(38\) 6.43303i 1.04358i
\(39\) 0 0
\(40\) 0 0
\(41\) −2.19039 −0.342082 −0.171041 0.985264i \(-0.554713\pi\)
−0.171041 + 0.985264i \(0.554713\pi\)
\(42\) 0 0
\(43\) 0.954564i 0.145570i 0.997348 + 0.0727849i \(0.0231886\pi\)
−0.997348 + 0.0727849i \(0.976811\pi\)
\(44\) 5.28808i 0.797207i
\(45\) 0 0
\(46\) −7.47847 −1.10264
\(47\) 11.0092i 1.60585i 0.596077 + 0.802927i \(0.296725\pi\)
−0.596077 + 0.802927i \(0.703275\pi\)
\(48\) 0 0
\(49\) 6.98188 + 0.503406i 0.997411 + 0.0719152i
\(50\) 0 0
\(51\) 0 0
\(52\) 2.19039 0.303753
\(53\) −3.09768 −0.425500 −0.212750 0.977107i \(-0.568242\pi\)
−0.212750 + 0.977107i \(0.568242\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.64404 0.0951965i −0.353324 0.0127212i
\(57\) 0 0
\(58\) 7.47847i 0.981971i
\(59\) 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 \(0.827609\pi\)
\(62\) 9.09768i 1.15541i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 5.33535i 0.651817i 0.945401 + 0.325908i \(0.105670\pi\)
−0.945401 + 0.325908i \(0.894330\pi\)
\(68\) 1.04544i 0.126778i
\(69\) 0 0
\(70\) 0 0
\(71\) 6.43303i 0.763461i −0.924274 0.381730i \(-0.875328\pi\)
0.924274 0.381730i \(-0.124672\pi\)
\(72\) 0 0
\(73\) 4.57615 0.535598 0.267799 0.963475i \(-0.413704\pi\)
0.267799 + 0.963475i \(0.413704\pi\)
\(74\) 0.855043i 0.0993967i
\(75\) 0 0
\(76\) 6.43303i 0.737920i
\(77\) −0.503406 + 13.9819i −0.0573684 + 1.59338i
\(78\) 0 0
\(79\) −15.6738 −1.76344 −0.881722 0.471769i \(-0.843616\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2.19039 0.241888
\(83\) 4.38079i 0.480854i −0.970667 0.240427i \(-0.922713\pi\)
0.970667 0.240427i \(-0.0772874\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.954564i 0.102933i
\(87\) 0 0
\(88\) 5.28808i 0.563711i
\(89\) 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
\(92\) 7.47847 0.779684
\(93\) 0 0
\(94\) 11.0092i 1.13551i
\(95\) 0 0
\(96\) 0 0
\(97\) 11.8593 1.20412 0.602062 0.798449i \(-0.294346\pi\)
0.602062 + 0.798449i \(0.294346\pi\)
\(98\) −6.98188 0.503406i −0.705276 0.0508517i
\(99\) 0 0
\(100\) 0 0
\(101\) −11.5830 −1.15255 −0.576274 0.817257i \(-0.695494\pi\)
−0.576274 + 0.817257i \(0.695494\pi\)
\(102\) 0 0
\(103\) 16.6757 1.64310 0.821552 0.570134i \(-0.193109\pi\)
0.821552 + 0.570134i \(0.193109\pi\)
\(104\) −2.19039 −0.214786
\(105\) 0 0
\(106\) 3.09768 0.300874
\(107\) −7.00681 −0.677374 −0.338687 0.940899i \(-0.609983\pi\)
−0.338687 + 0.940899i \(0.609983\pi\)
\(108\) 0 0
\(109\) 4.28991 0.410899 0.205450 0.978668i \(-0.434134\pi\)
0.205450 + 0.978668i \(0.434134\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.64404 + 0.0951965i 0.249838 + 0.00899522i
\(113\) −8.09087 −0.761125 −0.380563 0.924755i \(-0.624270\pi\)
−0.380563 + 0.924755i \(0.624270\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 7.47847i 0.694358i
\(117\) 0 0
\(118\) −13.7734 −1.26794
\(119\) −0.0995218 + 2.76417i −0.00912314 + 0.253391i
\(120\) 0 0
\(121\) −16.9638 −1.54216
\(122\) 8.05225i 0.729016i
\(123\) 0 0
\(124\) 9.09768i 0.816996i
\(125\) 0 0
\(126\) 0 0
\(127\) 8.57118i 0.760569i 0.924870 + 0.380285i \(0.124174\pi\)
−0.924870 + 0.380285i \(0.875826\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −5.90048 −0.515527 −0.257764 0.966208i \(-0.582986\pi\)
−0.257764 + 0.966208i \(0.582986\pi\)
\(132\) 0 0
\(133\) 0.612402 17.0092i 0.0531020 1.47488i
\(134\) 5.33535i 0.460904i
\(135\) 0 0
\(136\) 1.04544i 0.0896454i
\(137\) 6.86607 0.586608 0.293304 0.956019i \(-0.405245\pi\)
0.293304 + 0.956019i \(0.405245\pi\)
\(138\) 0 0
\(139\) 13.9115i 1.17996i −0.807418 0.589979i \(-0.799136\pi\)
0.807418 0.589979i \(-0.200864\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.43303i 0.539848i
\(143\) 11.5830i 0.968616i
\(144\) 0 0
\(145\) 0 0
\(146\) −4.57615 −0.378725
\(147\) 0 0
\(148\) 0.855043i 0.0702841i
\(149\) 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 \(0.273404\pi\)
\(152\) 6.43303i 0.521788i
\(153\) 0 0
\(154\) 0.503406 13.9819i 0.0405656 1.12669i
\(155\) 0 0
\(156\) 0 0
\(157\) 24.7665 1.97659 0.988293 0.152569i \(-0.0487548\pi\)
0.988293 + 0.152569i \(0.0487548\pi\)
\(158\) 15.6738 1.24694
\(159\) 0 0
\(160\) 0 0
\(161\) 19.7734 + 0.711924i 1.55836 + 0.0561075i
\(162\) 0 0
\(163\) 7.42622i 0.581667i −0.956774 0.290833i \(-0.906068\pi\)
0.956774 0.290833i \(-0.0939325\pi\)
\(164\) −2.19039 −0.171041
\(165\) 0 0
\(166\) 4.38079i 0.340015i
\(167\) 0.573779i 0.0444003i −0.999754 0.0222002i \(-0.992933\pi\)
0.999754 0.0222002i \(-0.00706711\pi\)
\(168\) 0 0
\(169\) −8.20218 −0.630937
\(170\) 0 0
\(171\) 0 0
\(172\) 0.954564i 0.0727849i
\(173\) 9.96375i 0.757530i −0.925493 0.378765i \(-0.876349\pi\)
0.925493 0.378765i \(-0.123651\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.28808i 0.398604i
\(177\) 0 0
\(178\) −4.28126 −0.320894
\(179\) 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 \(0.865824\pi\)
\(182\) −5.79148 0.208518i −0.429293 0.0154564i
\(183\) 0 0
\(184\) −7.47847 −0.551320
\(185\) 0 0
\(186\) 0 0
\(187\) −5.52834 −0.404272
\(188\) 11.0092i 0.802927i
\(189\) 0 0
\(190\) 0 0
\(191\) 5.56697i 0.402812i −0.979508 0.201406i \(-0.935449\pi\)
0.979508 0.201406i \(-0.0645510\pi\)
\(192\) 0 0
\(193\) 8.47166i 0.609803i 0.952384 + 0.304902i \(0.0986236\pi\)
−0.952384 + 0.304902i \(0.901376\pi\)
\(194\) −11.8593 −0.851445
\(195\) 0 0
\(196\) 6.98188 + 0.503406i 0.498705 + 0.0359576i
\(197\) 12.8661 0.916669 0.458335 0.888780i \(-0.348446\pi\)
0.458335 + 0.888780i \(0.348446\pi\)
\(198\) 0 0
\(199\) 17.5830i 1.24642i 0.782053 + 0.623212i \(0.214172\pi\)
−0.782053 + 0.623212i \(0.785828\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 11.5830 0.814975
\(203\) 0.711924 19.7734i 0.0499673 1.38782i
\(204\) 0 0
\(205\) 0 0
\(206\) −16.6757 −1.16185
\(207\) 0 0
\(208\) 2.19039 0.151876
\(209\) 34.0184 2.35310
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −3.09768 −0.212750
\(213\) 0 0
\(214\) 7.00681 0.478976
\(215\) 0 0
\(216\) 0 0
\(217\) 0.866067 24.0546i 0.0587925 1.63293i
\(218\) −4.28991 −0.290550
\(219\) 0 0
\(220\) 0 0
\(221\) 2.28991i 0.154036i
\(222\) 0 0
\(223\) 10.8710 0.727979 0.363989 0.931403i \(-0.381414\pi\)
0.363989 + 0.931403i \(0.381414\pi\)
\(224\) −2.64404 0.0951965i −0.176662 0.00636058i
\(225\) 0 0
\(226\) 8.09087 0.538197
\(227\) 14.0909i 0.935244i 0.883929 + 0.467622i \(0.154889\pi\)
−0.883929 + 0.467622i \(0.845111\pi\)
\(228\) 0 0
\(229\) 14.5239i 0.959767i −0.877332 0.479883i \(-0.840679\pi\)
0.877332 0.479883i \(-0.159321\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 7.47847i 0.490986i
\(233\) −20.6806 −1.35483 −0.677417 0.735599i \(-0.736901\pi\)
−0.677417 + 0.735599i \(0.736901\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 13.7734 0.896569
\(237\) 0 0
\(238\) 0.0995218 2.76417i 0.00645104 0.179175i
\(239\) 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 \(0.880504\pi\)
\(242\) 16.9638 1.09047
\(243\) 0 0
\(244\) 8.05225i 0.515492i
\(245\) 0 0
\(246\) 0 0
\(247\) 14.0909i 0.896581i
\(248\) 9.09768i 0.577703i
\(249\) 0 0
\(250\) 0 0
\(251\) −1.71874 −0.108486 −0.0542428 0.998528i \(-0.517275\pi\)
−0.0542428 + 0.998528i \(0.517275\pi\)
\(252\) 0 0
\(253\) 39.5467i 2.48628i
\(254\) 8.57118i 0.537804i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.18779i 0.136471i −0.997669 0.0682354i \(-0.978263\pi\)
0.997669 0.0682354i \(-0.0217369\pi\)
\(258\) 0 0
\(259\) 0.0813970 2.26077i 0.00505777 0.140477i
\(260\) 0 0
\(261\) 0 0
\(262\) 5.90048 0.364533
\(263\) 17.3876 1.07217 0.536083 0.844166i \(-0.319904\pi\)
0.536083 + 0.844166i \(0.319904\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.612402 + 17.0092i −0.0375488 + 1.04290i
\(267\) 0 0
\(268\) 5.33535i 0.325908i
\(269\) 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 \(0.985701\pi\)
\(272\) 1.04544i 0.0633888i
\(273\) 0 0
\(274\) −6.86607 −0.414794
\(275\) 0 0
\(276\) 0 0
\(277\) 10.7642i 0.646756i 0.946270 + 0.323378i \(0.104819\pi\)
−0.946270 + 0.323378i \(0.895181\pi\)
\(278\) 13.9115i 0.834356i
\(279\) 0 0
\(280\) 0 0
\(281\) 2.42806i 0.144846i 0.997374 + 0.0724230i \(0.0230731\pi\)
−0.997374 + 0.0724230i \(0.976927\pi\)
\(282\) 0 0
\(283\) −24.5898 −1.46171 −0.730855 0.682532i \(-0.760879\pi\)
−0.730855 + 0.682532i \(0.760879\pi\)
\(284\) 6.43303i 0.381730i
\(285\) 0 0
\(286\) 11.5830i 0.684915i
\(287\) −5.79148 0.208518i −0.341860 0.0123084i
\(288\) 0 0
\(289\) 15.9071 0.935710
\(290\) 0 0
\(291\) 0 0
\(292\) 4.57615 0.267799
\(293\) 12.6707i 0.740230i 0.928986 + 0.370115i \(0.120682\pi\)
−0.928986 + 0.370115i \(0.879318\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.855043i 0.0496983i
\(297\) 0 0
\(298\) 15.9638i 0.924755i
\(299\) 16.3808 0.947325
\(300\) 0 0
\(301\) −0.0908711 + 2.52390i −0.00523773 + 0.145475i
\(302\) −16.0546 −0.923840
\(303\) 0 0
\(304\) 6.43303i 0.368960i
\(305\) 0 0
\(306\) 0 0
\(307\) −0.866067 −0.0494291 −0.0247145 0.999695i \(-0.507868\pi\)
−0.0247145 + 0.999695i \(0.507868\pi\)
\(308\) −0.503406 + 13.9819i −0.0286842 + 0.796691i
\(309\) 0 0
\(310\) 0 0
\(311\) 9.71009 0.550608 0.275304 0.961357i \(-0.411221\pi\)
0.275304 + 0.961357i \(0.411221\pi\)
\(312\) 0 0
\(313\) 5.80096 0.327889 0.163945 0.986470i \(-0.447578\pi\)
0.163945 + 0.986470i \(0.447578\pi\)
\(314\) −24.7665 −1.39766
\(315\) 0 0
\(316\) −15.6738 −0.881722
\(317\) −28.1591 −1.58157 −0.790787 0.612092i \(-0.790328\pi\)
−0.790787 + 0.612092i \(0.790328\pi\)
\(318\) 0 0
\(319\) 39.5467 2.21419
\(320\) 0 0
\(321\) 0 0
\(322\) −19.7734 0.711924i −1.10193 0.0396740i
\(323\) 6.72532 0.374207
\(324\) 0 0
\(325\) 0 0
\(326\) 7.42622i 0.411300i
\(327\) 0 0
\(328\) 2.19039 0.120944
\(329\) −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 \(0.680891\pi\)
\(332\) 4.38079i 0.240427i
\(333\) 0 0
\(334\) 0.573779i 0.0313958i
\(335\) 0 0
\(336\) 0 0
\(337\) 28.3992i 1.54700i −0.633796 0.773500i \(-0.718504\pi\)
0.633796 0.773500i \(-0.281496\pi\)
\(338\) 8.20218 0.446140
\(339\) 0 0
\(340\) 0 0
\(341\) 48.1092 2.60526
\(342\) 0 0
\(343\) 18.4124 + 1.99567i 0.994177 + 0.107756i
\(344\) 0.954564i 0.0514667i
\(345\) 0 0
\(346\) 9.96375i 0.535655i
\(347\) 12.5898 0.675855 0.337927 0.941172i \(-0.390274\pi\)
0.337927 + 0.941172i \(0.390274\pi\)
\(348\) 0 0
\(349\) 20.9956i 1.12387i 0.827183 + 0.561933i \(0.189942\pi\)
−0.827183 + 0.561933i \(0.810058\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.28808i 0.281855i
\(353\) 15.1363i 0.805624i −0.915283 0.402812i \(-0.868033\pi\)
0.915283 0.402812i \(-0.131967\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 4.28126 0.226907
\(357\) 0 0
\(358\) 1.77336i 0.0937249i
\(359\) 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
\(362\) 11.0092i 0.578630i
\(363\) 0 0
\(364\) 5.79148 + 0.208518i 0.303556 + 0.0109293i
\(365\) 0 0
\(366\) 0 0
\(367\) 11.6825 0.609821 0.304910 0.952381i \(-0.401373\pi\)
0.304910 + 0.952381i \(0.401373\pi\)
\(368\) 7.47847 0.389842
\(369\) 0 0
\(370\) 0 0
\(371\) −8.19039 0.294888i −0.425224 0.0153098i
\(372\) 0 0
\(373\) 12.8550i 0.665609i −0.942996 0.332804i \(-0.892005\pi\)
0.942996 0.332804i \(-0.107995\pi\)
\(374\) 5.52834 0.285864
\(375\) 0 0
\(376\) 11.0092i 0.567755i
\(377\) 16.3808i 0.843653i
\(378\) 0 0
\(379\) −25.0751 −1.28802 −0.644010 0.765017i \(-0.722730\pi\)
−0.644010 + 0.765017i \(0.722730\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 5.56697i 0.284831i
\(383\) 23.0092i 1.17571i −0.808965 0.587857i \(-0.799972\pi\)
0.808965 0.587857i \(-0.200028\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.47166i 0.431196i
\(387\) 0 0
\(388\) 11.8593 0.602062
\(389\) 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
\(392\) −6.98188 0.503406i −0.352638 0.0254258i
\(393\) 0 0
\(394\) −12.8661 −0.648183
\(395\) 0 0
\(396\) 0 0
\(397\) 13.9141 0.698329 0.349164 0.937061i \(-0.386465\pi\)
0.349164 + 0.937061i \(0.386465\pi\)
\(398\) 17.5830i 0.881354i
\(399\) 0 0
\(400\) 0 0
\(401\) 14.6197i 0.730075i −0.930993 0.365038i \(-0.881056\pi\)
0.930993 0.365038i \(-0.118944\pi\)
\(402\) 0 0
\(403\) 19.9275i 0.992660i
\(404\) −11.5830 −0.576274
\(405\) 0 0
\(406\) −0.711924 + 19.7734i −0.0353322 + 0.981335i
\(407\) 4.52153 0.224124
\(408\) 0 0
\(409\) 12.0000i 0.593362i 0.954977 + 0.296681i \(0.0958798\pi\)
−0.954977 + 0.296681i \(0.904120\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 16.6757 0.821552
\(413\) 36.4173 + 1.31117i 1.79198 + 0.0645187i
\(414\) 0 0
\(415\) 0 0
\(416\) −2.19039 −0.107393
\(417\) 0 0
\(418\) −34.0184 −1.66389
\(419\) 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 \(0.493387\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) 3.09768 0.150437
\(425\) 0 0
\(426\) 0 0
\(427\) −0.766545 + 21.2905i −0.0370957 + 1.03032i
\(428\) −7.00681 −0.338687
\(429\) 0 0
\(430\) 0 0
\(431\) 20.2476i 0.975293i 0.873041 + 0.487647i \(0.162145\pi\)
−0.873041 + 0.487647i \(0.837855\pi\)
\(432\) 0 0
\(433\) 30.6444 1.47268 0.736338 0.676614i \(-0.236553\pi\)
0.736338 + 0.676614i \(0.236553\pi\)
\(434\) −0.866067 + 24.0546i −0.0415726 + 1.15466i
\(435\) 0 0
\(436\) 4.28991 0.205450
\(437\) 48.1092i 2.30138i
\(438\) 0 0
\(439\) 22.6308i 1.08011i 0.841630 + 0.540054i \(0.181596\pi\)
−0.841630 + 0.540054i \(0.818404\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.28991i 0.108920i
\(443\) −36.3083 −1.72506 −0.862529 0.506007i \(-0.831121\pi\)
−0.862529 + 0.506007i \(0.831121\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −10.8710 −0.514759
\(447\) 0 0
\(448\) 2.64404 + 0.0951965i 0.124919 + 0.00449761i
\(449\) 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
\(452\) −8.09087 −0.380563
\(453\) 0 0
\(454\) 14.0909i 0.661317i
\(455\) 0 0
\(456\) 0 0
\(457\) 25.3477i 1.18571i 0.805308 + 0.592857i \(0.202000\pi\)
−0.805308 + 0.592857i \(0.798000\pi\)
\(458\) 14.5239i 0.678657i
\(459\) 0 0
\(460\) 0 0
\(461\) 12.7253 0.592677 0.296339 0.955083i \(-0.404234\pi\)
0.296339 + 0.955083i \(0.404234\pi\)
\(462\) 0 0
\(463\) 21.2655i 0.988289i 0.869380 + 0.494145i \(0.164519\pi\)
−0.869380 + 0.494145i \(0.835481\pi\)
\(464\) 7.47847i 0.347179i
\(465\) 0 0
\(466\) 20.6806 0.958013
\(467\) 13.1424i 0.608156i 0.952647 + 0.304078i \(0.0983483\pi\)
−0.952647 + 0.304078i \(0.901652\pi\)
\(468\) 0 0
\(469\) −0.507906 + 14.1069i −0.0234529 + 0.651395i
\(470\) 0 0
\(471\) 0 0
\(472\) −13.7734 −0.633970
\(473\) −5.04781 −0.232099
\(474\) 0 0
\(475\) 0 0
\(476\) −0.0995218 + 2.76417i −0.00456157 + 0.126696i
\(477\) 0 0
\(478\) 5.56697i 0.254627i
\(479\) −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
\(482\) 11.3839i 0.518523i
\(483\) 0 0
\(484\) −16.9638 −0.771080
\(485\) 0 0
\(486\) 0 0
\(487\) 39.5381i 1.79164i 0.444416 + 0.895820i \(0.353411\pi\)
−0.444416 + 0.895820i \(0.646589\pi\)
\(488\) 8.05225i 0.364508i
\(489\) 0 0
\(490\) 0 0
\(491\) 4.62105i 0.208545i −0.994549 0.104273i \(-0.966749\pi\)
0.994549 0.104273i \(-0.0332514\pi\)
\(492\) 0 0
\(493\) 7.81826 0.352117
\(494\) 14.0909i 0.633978i
\(495\) 0 0
\(496\) 9.09768i 0.408498i
\(497\) 0.612402 17.0092i 0.0274700 0.762966i
\(498\) 0 0
\(499\) −31.0152 −1.38843 −0.694216 0.719766i \(-0.744249\pi\)
−0.694216 + 0.719766i \(0.744249\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1.71874 0.0767109
\(503\) 28.3385i 1.26355i 0.775152 + 0.631775i \(0.217673\pi\)
−0.775152 + 0.631775i \(0.782327\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 39.5467i 1.75807i
\(507\) 0 0
\(508\) 8.57118i 0.380285i
\(509\) −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
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 2.18779i 0.0964994i
\(515\) 0 0
\(516\) 0 0
\(517\) −58.2174 −2.56040
\(518\) −0.0813970 + 2.26077i −0.00357638 + 0.0993323i
\(519\) 0 0
\(520\) 0 0
\(521\) 28.5896 1.25253 0.626265 0.779610i \(-0.284583\pi\)
0.626265 + 0.779610i \(0.284583\pi\)
\(522\) 0 0
\(523\) −3.51472 −0.153688 −0.0768440 0.997043i \(-0.524484\pi\)
−0.0768440 + 0.997043i \(0.524484\pi\)
\(524\) −5.90048 −0.257764
\(525\) 0 0
\(526\) −17.3876 −0.758135
\(527\) 9.51104 0.414307
\(528\) 0 0
\(529\) 32.9275 1.43163
\(530\) 0 0
\(531\) 0 0
\(532\) 0.612402 17.0092i 0.0265510 0.737442i
\(533\) −4.79782 −0.207817
\(534\) 0 0
\(535\) 0 0
\(536\) 5.33535i 0.230452i
\(537\) 0 0
\(538\) −20.3445 −0.877115
\(539\) −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 \(0.272486\pi\)
\(542\) 1.47847i 0.0635057i
\(543\) 0 0
\(544\) 1.04544i 0.0448227i
\(545\) 0 0
\(546\) 0 0
\(547\) 6.86369i 0.293470i −0.989176 0.146735i \(-0.953123\pi\)
0.989176 0.146735i \(-0.0468765\pi\)
\(548\) 6.86607 0.293304
\(549\) 0 0
\(550\) 0 0
\(551\) −48.1092 −2.04952
\(552\) 0 0
\(553\) −41.4422 1.49209i −1.76230 0.0634503i
\(554\) 10.7642i 0.457326i
\(555\) 0 0
\(556\) 13.9115i 0.589979i
\(557\) 1.81826 0.0770421 0.0385210 0.999258i \(-0.487735\pi\)
0.0385210 + 0.999258i \(0.487735\pi\)
\(558\) 0 0
\(559\) 2.09087i 0.0884344i
\(560\) 0 0
\(561\) 0 0
\(562\) 2.42806i 0.102422i
\(563\) 34.0184i 1.43370i −0.697226 0.716852i \(-0.745582\pi\)
0.697226 0.716852i \(-0.254418\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 24.5898 1.03359
\(567\) 0 0
\(568\) 6.43303i 0.269924i
\(569\) 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 \(0.427294\pi\)
\(572\) 11.5830i 0.484308i
\(573\) 0 0
\(574\) 5.79148 + 0.208518i 0.241732 + 0.00870336i
\(575\) 0 0
\(576\) 0 0
\(577\) −39.1523 −1.62993 −0.814966 0.579509i \(-0.803244\pi\)
−0.814966 + 0.579509i \(0.803244\pi\)
\(578\) −15.9071 −0.661647
\(579\) 0 0
\(580\) 0 0
\(581\) 0.417035 11.5830i 0.0173015 0.480542i
\(582\) 0 0
\(583\) 16.3808i 0.678423i
\(584\) −4.57615 −0.189363
\(585\) 0 0
\(586\) 12.6707i 0.523422i
\(587\) 5.52834i 0.228179i 0.993470 + 0.114090i \(0.0363951\pi\)
−0.993470 + 0.114090i \(0.963605\pi\)
\(588\) 0 0
\(589\) −58.5257 −2.41151
\(590\) 0 0
\(591\) 0 0
\(592\) 0.855043i 0.0351420i
\(593\) 18.6830i 0.767220i 0.923495 + 0.383610i \(0.125319\pi\)
−0.923495 + 0.383610i \(0.874681\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.9638i 0.653901i
\(597\) 0 0
\(598\) −16.3808 −0.669860
\(599\) 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 \(0.828811\pi\)
\(602\) 0.0908711 2.52390i 0.00370363 0.102867i
\(603\) 0 0
\(604\) 16.0546 0.653253
\(605\) 0 0
\(606\) 0 0
\(607\) −31.4336 −1.27585 −0.637925 0.770099i \(-0.720207\pi\)
−0.637925 + 0.770099i \(0.720207\pi\)
\(608\) 6.43303i 0.260894i
\(609\) 0 0
\(610\) 0 0
\(611\) 24.1144i 0.975566i
\(612\) 0 0
\(613\) 39.4532i 1.59350i −0.604308 0.796751i \(-0.706550\pi\)
0.604308 0.796751i \(-0.293450\pi\)
\(614\) 0.866067 0.0349516
\(615\) 0 0
\(616\) 0.503406 13.9819i 0.0202828 0.563346i
\(617\) 39.7421 1.59996 0.799978 0.600029i \(-0.204844\pi\)
0.799978 + 0.600029i \(0.204844\pi\)
\(618\) 0 0
\(619\) 14.7776i 0.593961i 0.954884 + 0.296980i \(0.0959796\pi\)
−0.954884 + 0.296980i \(0.904020\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −9.71009 −0.389339
\(623\) 11.3198 + 0.407561i 0.453519 + 0.0163286i
\(624\) 0 0
\(625\) 0 0
\(626\) −5.80096 −0.231853
\(627\) 0 0
\(628\) 24.7665 0.988293
\(629\) 0.893892 0.0356418
\(630\) 0 0
\(631\) −16.3083 −0.649223 −0.324611 0.945847i \(-0.605233\pi\)
−0.324611 + 0.945847i \(0.605233\pi\)
\(632\) 15.6738 0.623472
\(633\) 0 0
\(634\) 28.1591 1.11834
\(635\) 0 0
\(636\) 0 0
\(637\) 15.2931 + 1.10266i 0.605933 + 0.0436889i
\(638\) −39.5467 −1.56567
\(639\) 0 0
\(640\) 0 0
\(641\) 12.5289i 0.494861i −0.968906 0.247430i \(-0.920414\pi\)
0.968906 0.247430i \(-0.0795862\pi\)
\(642\) 0 0
\(643\) −14.3992 −0.567847 −0.283924 0.958847i \(-0.591636\pi\)
−0.283924 + 0.958847i \(0.591636\pi\)
\(644\) 19.7734 + 0.711924i 0.779179 + 0.0280537i
\(645\) 0 0
\(646\) −6.72532 −0.264604
\(647\) 31.7708i 1.24904i 0.781010 + 0.624519i \(0.214705\pi\)
−0.781010 + 0.624519i \(0.785295\pi\)
\(648\) 0 0
\(649\) 72.8346i 2.85901i
\(650\) 0 0
\(651\) 0 0
\(652\) 7.42622i 0.290833i
\(653\) −20.7389 −0.811578 −0.405789 0.913967i \(-0.633003\pi\)
−0.405789 + 0.913967i \(0.633003\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.19039 −0.0855205
\(657\) 0 0
\(658\) 1.04804 29.1087i 0.0408567 1.13478i
\(659\) 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 \(-0.224933\pi\)
\(662\) 19.5830 0.761114
\(663\) 0 0
\(664\) 4.38079i 0.170007i
\(665\) 0 0
\(666\) 0 0
\(667\) 55.9275i 2.16552i
\(668\) 0.573779i 0.0222002i
\(669\) 0 0
\(670\) 0 0
\(671\) −42.5809 −1.64382
\(672\) 0 0
\(673\) 18.6943i 0.720611i 0.932834 + 0.360306i \(0.117328\pi\)
−0.932834 + 0.360306i \(0.882672\pi\)
\(674\) 28.3992i 1.09389i
\(675\) 0 0
\(676\) −8.20218 −0.315468
\(677\) 36.5861i 1.40612i −0.711131 0.703059i \(-0.751817\pi\)
0.711131 0.703059i \(-0.248183\pi\)
\(678\) 0 0
\(679\) 31.3563 + 1.12896i 1.20335 + 0.0433255i
\(680\) 0 0
\(681\) 0 0
\(682\) −48.1092 −1.84220
\(683\) 36.9207 1.41273 0.706365 0.707847i \(-0.250334\pi\)
0.706365 + 0.707847i \(0.250334\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −18.4124 1.99567i −0.702990 0.0761952i
\(687\) 0 0
\(688\) 0.954564i 0.0363924i
\(689\) −6.78514 −0.258493
\(690\) 0 0
\(691\) 5.20823i 0.198130i 0.995081 + 0.0990652i \(0.0315852\pi\)
−0.995081 + 0.0990652i \(0.968415\pi\)
\(692\) 9.96375i 0.378765i
\(693\) 0 0
\(694\) −12.5898 −0.477901
\(695\) 0 0
\(696\) 0 0
\(697\) 2.28991i 0.0867367i
\(698\) 20.9956i 0.794694i
\(699\) 0 0
\(700\) 0 0
\(701\) 13.2831i 0.501696i −0.968027 0.250848i \(-0.919291\pi\)
0.968027 0.250848i \(-0.0807094\pi\)
\(702\) 0 0
\(703\) −5.50052 −0.207456
\(704\) 5.28808i 0.199302i
\(705\) 0 0
\(706\) 15.1363i 0.569662i
\(707\) −30.6258 1.10266i −1.15180 0.0414697i
\(708\) 0 0
\(709\) 4.09087 0.153636 0.0768179 0.997045i \(-0.475524\pi\)
0.0768179 + 0.997045i \(0.475524\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −4.28126 −0.160447
\(713\) 68.0367i 2.54800i
\(714\) 0 0
\(715\) 0 0
\(716\) 1.77336i 0.0662735i
\(717\) 0 0
\(718\) 33.1909i 1.23867i
\(719\) −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
\(722\) 22.3839 0.833043
\(723\) 0 0
\(724\) 11.0092i 0.409153i
\(725\) 0 0
\(726\) 0 0
\(727\) −40.7849 −1.51263 −0.756314 0.654208i \(-0.773002\pi\)
−0.756314 + 0.654208i \(0.773002\pi\)
\(728\) −5.79148 0.208518i −0.214647 0.00772818i
\(729\) 0 0
\(730\) 0 0
\(731\) −0.997936 −0.0369100
\(732\) 0 0
\(733\) −39.2481 −1.44966 −0.724832 0.688926i \(-0.758083\pi\)
−0.724832 + 0.688926i \(0.758083\pi\)
\(734\) −11.6825 −0.431208
\(735\) 0 0
\(736\) −7.47847 −0.275660
\(737\) −28.2137 −1.03927
\(738\) 0 0
\(739\) 16.7305 0.615442 0.307721 0.951477i \(-0.400434\pi\)
0.307721 + 0.951477i \(0.400434\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 8.19039 + 0.294888i 0.300679 + 0.0108257i
\(743\) −35.7185 −1.31039 −0.655193 0.755462i \(-0.727412\pi\)
−0.655193 + 0.755462i \(0.727412\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 12.8550i 0.470657i
\(747\) 0 0
\(748\) −5.52834 −0.202136
\(749\) −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 \(0.553345\pi\)
\(752\) 11.0092i 0.401464i
\(753\) 0 0
\(754\) 16.3808i 0.596553i
\(755\) 0 0
\(756\) 0 0
\(757\) 48.2027i 1.75196i 0.482350 + 0.875979i \(0.339783\pi\)
−0.482350 + 0.875979i \(0.660217\pi\)
\(758\) 25.0751 0.910767
\(759\) 0 0
\(760\) 0 0
\(761\) 9.61056 0.348383 0.174191 0.984712i \(-0.444269\pi\)
0.174191 + 0.984712i \(0.444269\pi\)
\(762\) 0 0
\(763\) 11.3427 + 0.408385i 0.410633 + 0.0147845i
\(764\) 5.56697i 0.201406i
\(765\) 0 0
\(766\) 23.0092i 0.831356i
\(767\) 30.1691 1.08934
\(768\) 0 0
\(769\) 33.2931i 1.20058i −0.799783 0.600289i \(-0.795052\pi\)
0.799783 0.600289i \(-0.204948\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.47166i 0.304902i
\(773\) 12.2726i 0.441415i 0.975340 + 0.220708i \(0.0708367\pi\)
−0.975340 + 0.220708i \(0.929163\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −11.8593 −0.425722
\(777\) 0 0
\(778\) 10.1591i 0.364222i
\(779\) 14.0909i 0.504858i
\(780\) 0 0
\(781\) 34.0184 1.21727
\(782\) 7.81826i 0.279580i
\(783\) 0 0
\(784\) 6.98188 + 0.503406i 0.249353 + 0.0179788i
\(785\) 0 0
\(786\) 0 0
\(787\) −21.4286 −0.763847 −0.381923 0.924194i \(-0.624738\pi\)
−0.381923 + 0.924194i \(0.624738\pi\)
\(788\) 12.8661 0.458335
\(789\) 0 0
\(790\) 0 0
\(791\) −21.3926 0.770222i −0.760632 0.0273859i
\(792\) 0 0
\(793\) 17.6376i 0.626329i
\(794\) −13.9141 −0.493793
\(795\) 0 0
\(796\) 17.5830i 0.623212i
\(797\) 9.96375i 0.352934i −0.984307 0.176467i \(-0.943533\pi\)
0.984307 0.176467i \(-0.0564669\pi\)
\(798\) 0 0
\(799\) −11.5094 −0.407173
\(800\) 0 0
\(801\) 0 0
\(802\) 14.6197i 0.516241i