Properties

Label 3150.2.d.b.3149.5
Level 3150
Weight 2
Character 3150.3149
Analytic conductor 25.153
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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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: 8.0.40960000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3149.5
Root \(-0.437016 + 0.437016i\)
Character \(\chi\) = 3150.3149
Dual form 3150.2.d.b.3149.7

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +(2.23607 - 1.41421i) q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +(2.23607 - 1.41421i) q^{7} -1.00000 q^{8} -1.41421i q^{11} -0.926210 q^{13} +(-2.23607 + 1.41421i) q^{14} +1.00000 q^{16} +2.23607i q^{17} -7.63441i q^{19} +1.41421i q^{22} -1.00000 q^{23} +0.926210 q^{26} +(2.23607 - 1.41421i) q^{28} +0.757359i q^{29} -4.08849i q^{31} -1.00000 q^{32} -2.23607i q^{34} +2.82843i q^{37} +7.63441i q^{38} -8.56062 q^{41} -3.58579i q^{43} -1.41421i q^{44} +1.00000 q^{46} -1.30986i q^{47} +(3.00000 - 6.32456i) q^{49} -0.926210 q^{52} -8.07107 q^{53} +(-2.23607 + 1.41421i) q^{56} -0.757359i q^{58} +7.25077 q^{59} -0.926210i q^{61} +4.08849i q^{62} +1.00000 q^{64} +2.23607i q^{68} -15.6569i q^{71} -13.9590 q^{73} -2.82843i q^{74} -7.63441i q^{76} +(-2.00000 - 3.16228i) q^{77} +13.0711 q^{79} +8.56062 q^{82} +14.3426i q^{83} +3.58579i q^{86} +1.41421i q^{88} -2.61972 q^{89} +(-2.07107 + 1.30986i) q^{91} -1.00000 q^{92} +1.30986i q^{94} +0.542561 q^{97} +(-3.00000 + 6.32456i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{2} + 8q^{4} - 8q^{8} + O(q^{10}) \) \( 8q - 8q^{2} + 8q^{4} - 8q^{8} + 8q^{16} - 8q^{23} - 8q^{32} + 8q^{46} + 24q^{49} - 8q^{53} + 8q^{64} - 16q^{77} + 48q^{79} + 40q^{91} - 8q^{92} - 24q^{98} + O(q^{100}) \)

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
\(7\) 2.23607 1.41421i 0.845154 0.534522i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421i 0.426401i −0.977008 0.213201i \(-0.931611\pi\)
0.977008 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) −0.926210 −0.256884 −0.128442 0.991717i \(-0.540998\pi\)
−0.128442 + 0.991717i \(0.540998\pi\)
\(14\) −2.23607 + 1.41421i −0.597614 + 0.377964i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.23607i 0.542326i 0.962533 + 0.271163i \(0.0874083\pi\)
−0.962533 + 0.271163i \(0.912592\pi\)
\(18\) 0 0
\(19\) 7.63441i 1.75145i −0.482806 0.875727i \(-0.660382\pi\)
0.482806 0.875727i \(-0.339618\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.41421i 0.301511i
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.926210 0.181645
\(27\) 0 0
\(28\) 2.23607 1.41421i 0.422577 0.267261i
\(29\) 0.757359i 0.140638i 0.997525 + 0.0703190i \(0.0224017\pi\)
−0.997525 + 0.0703190i \(0.977598\pi\)
\(30\) 0 0
\(31\) 4.08849i 0.734314i −0.930159 0.367157i \(-0.880331\pi\)
0.930159 0.367157i \(-0.119669\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.23607i 0.383482i
\(35\) 0 0
\(36\) 0 0
\(37\) 2.82843i 0.464991i 0.972598 + 0.232495i \(0.0746890\pi\)
−0.972598 + 0.232495i \(0.925311\pi\)
\(38\) 7.63441i 1.23847i
\(39\) 0 0
\(40\) 0 0
\(41\) −8.56062 −1.33694 −0.668472 0.743737i \(-0.733052\pi\)
−0.668472 + 0.743737i \(0.733052\pi\)
\(42\) 0 0
\(43\) 3.58579i 0.546827i −0.961897 0.273414i \(-0.911847\pi\)
0.961897 0.273414i \(-0.0881528\pi\)
\(44\) 1.41421i 0.213201i
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 1.30986i 0.191062i −0.995426 0.0955312i \(-0.969545\pi\)
0.995426 0.0955312i \(-0.0304550\pi\)
\(48\) 0 0
\(49\) 3.00000 6.32456i 0.428571 0.903508i
\(50\) 0 0
\(51\) 0 0
\(52\) −0.926210 −0.128442
\(53\) −8.07107 −1.10865 −0.554323 0.832301i \(-0.687023\pi\)
−0.554323 + 0.832301i \(0.687023\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.23607 + 1.41421i −0.298807 + 0.188982i
\(57\) 0 0
\(58\) 0.757359i 0.0994461i
\(59\) 7.25077 0.943969 0.471985 0.881607i \(-0.343538\pi\)
0.471985 + 0.881607i \(0.343538\pi\)
\(60\) 0 0
\(61\) 0.926210i 0.118589i −0.998241 0.0592945i \(-0.981115\pi\)
0.998241 0.0592945i \(-0.0188851\pi\)
\(62\) 4.08849i 0.519238i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 2.23607i 0.271163i
\(69\) 0 0
\(70\) 0 0
\(71\) 15.6569i 1.85813i −0.369921 0.929063i \(-0.620615\pi\)
0.369921 0.929063i \(-0.379385\pi\)
\(72\) 0 0
\(73\) −13.9590 −1.63377 −0.816887 0.576798i \(-0.804302\pi\)
−0.816887 + 0.576798i \(0.804302\pi\)
\(74\) 2.82843i 0.328798i
\(75\) 0 0
\(76\) 7.63441i 0.875727i
\(77\) −2.00000 3.16228i −0.227921 0.360375i
\(78\) 0 0
\(79\) 13.0711 1.47061 0.735305 0.677736i \(-0.237039\pi\)
0.735305 + 0.677736i \(0.237039\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 8.56062 0.945363
\(83\) 14.3426i 1.57431i 0.616757 + 0.787153i \(0.288446\pi\)
−0.616757 + 0.787153i \(0.711554\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.58579i 0.386665i
\(87\) 0 0
\(88\) 1.41421i 0.150756i
\(89\) −2.61972 −0.277689 −0.138845 0.990314i \(-0.544339\pi\)
−0.138845 + 0.990314i \(0.544339\pi\)
\(90\) 0 0
\(91\) −2.07107 + 1.30986i −0.217107 + 0.137310i
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 1.30986i 0.135102i
\(95\) 0 0
\(96\) 0 0
\(97\) 0.542561 0.0550887 0.0275444 0.999621i \(-0.491231\pi\)
0.0275444 + 0.999621i \(0.491231\pi\)
\(98\) −3.00000 + 6.32456i −0.303046 + 0.638877i
\(99\) 0 0
\(100\) 0 0
\(101\) −12.1065 −1.20465 −0.602323 0.798252i \(-0.705758\pi\)
−0.602323 + 0.798252i \(0.705758\pi\)
\(102\) 0 0
\(103\) −10.4130 −1.02603 −0.513014 0.858380i \(-0.671471\pi\)
−0.513014 + 0.858380i \(0.671471\pi\)
\(104\) 0.926210 0.0908223
\(105\) 0 0
\(106\) 8.07107 0.783931
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) 7.07107 0.677285 0.338643 0.940915i \(-0.390032\pi\)
0.338643 + 0.940915i \(0.390032\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.23607 1.41421i 0.211289 0.133631i
\(113\) −1.07107 −0.100758 −0.0503788 0.998730i \(-0.516043\pi\)
−0.0503788 + 0.998730i \(0.516043\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.757359i 0.0703190i
\(117\) 0 0
\(118\) −7.25077 −0.667487
\(119\) 3.16228 + 5.00000i 0.289886 + 0.458349i
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 0.926210i 0.0838551i
\(123\) 0 0
\(124\) 4.08849i 0.367157i
\(125\) 0 0
\(126\) 0 0
\(127\) 10.0000i 0.887357i 0.896186 + 0.443678i \(0.146327\pi\)
−0.896186 + 0.443678i \(0.853673\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −12.6491 −1.10516 −0.552579 0.833461i \(-0.686356\pi\)
−0.552579 + 0.833461i \(0.686356\pi\)
\(132\) 0 0
\(133\) −10.7967 17.0711i −0.936192 1.48025i
\(134\) 0 0
\(135\) 0 0
\(136\) 2.23607i 0.191741i
\(137\) 12.1421 1.03737 0.518686 0.854965i \(-0.326421\pi\)
0.518686 + 0.854965i \(0.326421\pi\)
\(138\) 0 0
\(139\) 10.7967i 0.915763i −0.889013 0.457882i \(-0.848608\pi\)
0.889013 0.457882i \(-0.151392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 15.6569i 1.31389i
\(143\) 1.30986i 0.109536i
\(144\) 0 0
\(145\) 0 0
\(146\) 13.9590 1.15525
\(147\) 0 0
\(148\) 2.82843i 0.232495i
\(149\) 0.757359i 0.0620453i −0.999519 0.0310226i \(-0.990124\pi\)
0.999519 0.0310226i \(-0.00987640\pi\)
\(150\) 0 0
\(151\) −14.1421 −1.15087 −0.575435 0.817847i \(-0.695167\pi\)
−0.575435 + 0.817847i \(0.695167\pi\)
\(152\) 7.63441i 0.619233i
\(153\) 0 0
\(154\) 2.00000 + 3.16228i 0.161165 + 0.254824i
\(155\) 0 0
\(156\) 0 0
\(157\) 10.7967 0.861670 0.430835 0.902431i \(-0.358219\pi\)
0.430835 + 0.902431i \(0.358219\pi\)
\(158\) −13.0711 −1.03988
\(159\) 0 0
\(160\) 0 0
\(161\) −2.23607 + 1.41421i −0.176227 + 0.111456i
\(162\) 0 0
\(163\) 7.92893i 0.621042i −0.950567 0.310521i \(-0.899497\pi\)
0.950567 0.310521i \(-0.100503\pi\)
\(164\) −8.56062 −0.668472
\(165\) 0 0
\(166\) 14.3426i 1.11320i
\(167\) 12.1065i 0.936833i −0.883508 0.468416i \(-0.844825\pi\)
0.883508 0.468416i \(-0.155175\pi\)
\(168\) 0 0
\(169\) −12.1421 −0.934010
\(170\) 0 0
\(171\) 0 0
\(172\) 3.58579i 0.273414i
\(173\) 10.2541i 0.779607i 0.920898 + 0.389804i \(0.127457\pi\)
−0.920898 + 0.389804i \(0.872543\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.41421i 0.106600i
\(177\) 0 0
\(178\) 2.61972 0.196356
\(179\) 11.3137i 0.845626i −0.906217 0.422813i \(-0.861043\pi\)
0.906217 0.422813i \(-0.138957\pi\)
\(180\) 0 0
\(181\) 2.61972i 0.194722i 0.995249 + 0.0973610i \(0.0310401\pi\)
−0.995249 + 0.0973610i \(0.968960\pi\)
\(182\) 2.07107 1.30986i 0.153518 0.0970932i
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 3.16228 0.231249
\(188\) 1.30986i 0.0955312i
\(189\) 0 0
\(190\) 0 0
\(191\) 0.656854i 0.0475283i 0.999718 + 0.0237642i \(0.00756508\pi\)
−0.999718 + 0.0237642i \(0.992435\pi\)
\(192\) 0 0
\(193\) 8.48528i 0.610784i −0.952227 0.305392i \(-0.901213\pi\)
0.952227 0.305392i \(-0.0987875\pi\)
\(194\) −0.542561 −0.0389536
\(195\) 0 0
\(196\) 3.00000 6.32456i 0.214286 0.451754i
\(197\) 5.92893 0.422419 0.211209 0.977441i \(-0.432260\pi\)
0.211209 + 0.977441i \(0.432260\pi\)
\(198\) 0 0
\(199\) 21.5934i 1.53071i −0.643606 0.765357i \(-0.722562\pi\)
0.643606 0.765357i \(-0.277438\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 12.1065 0.851814
\(203\) 1.07107 + 1.69351i 0.0751742 + 0.118861i
\(204\) 0 0
\(205\) 0 0
\(206\) 10.4130 0.725511
\(207\) 0 0
\(208\) −0.926210 −0.0642211
\(209\) −10.7967 −0.746823
\(210\) 0 0
\(211\) 16.2132 1.11616 0.558081 0.829786i \(-0.311538\pi\)
0.558081 + 0.829786i \(0.311538\pi\)
\(212\) −8.07107 −0.554323
\(213\) 0 0
\(214\) −2.00000 −0.136717
\(215\) 0 0
\(216\) 0 0
\(217\) −5.78199 9.14214i −0.392507 0.620609i
\(218\) −7.07107 −0.478913
\(219\) 0 0
\(220\) 0 0
\(221\) 2.07107i 0.139315i
\(222\) 0 0
\(223\) −13.0328 −0.872738 −0.436369 0.899768i \(-0.643736\pi\)
−0.436369 + 0.899768i \(0.643736\pi\)
\(224\) −2.23607 + 1.41421i −0.149404 + 0.0944911i
\(225\) 0 0
\(226\) 1.07107 0.0712464
\(227\) 25.1393i 1.66855i −0.551345 0.834277i \(-0.685885\pi\)
0.551345 0.834277i \(-0.314115\pi\)
\(228\) 0 0
\(229\) 10.7967i 0.713465i 0.934206 + 0.356733i \(0.116109\pi\)
−0.934206 + 0.356733i \(0.883891\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.757359i 0.0497231i
\(233\) 23.0711 1.51144 0.755718 0.654897i \(-0.227288\pi\)
0.755718 + 0.654897i \(0.227288\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 7.25077 0.471985
\(237\) 0 0
\(238\) −3.16228 5.00000i −0.204980 0.324102i
\(239\) 7.17157i 0.463890i 0.972729 + 0.231945i \(0.0745090\pi\)
−0.972729 + 0.231945i \(0.925491\pi\)
\(240\) 0 0
\(241\) 4.47214i 0.288076i 0.989572 + 0.144038i \(0.0460087\pi\)
−0.989572 + 0.144038i \(0.953991\pi\)
\(242\) −9.00000 −0.578542
\(243\) 0 0
\(244\) 0.926210i 0.0592945i
\(245\) 0 0
\(246\) 0 0
\(247\) 7.07107i 0.449921i
\(248\) 4.08849i 0.259619i
\(249\) 0 0
\(250\) 0 0
\(251\) −18.8148 −1.18758 −0.593788 0.804621i \(-0.702368\pi\)
−0.593788 + 0.804621i \(0.702368\pi\)
\(252\) 0 0
\(253\) 1.41421i 0.0889108i
\(254\) 10.0000i 0.627456i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 21.2097i 1.32303i −0.749933 0.661513i \(-0.769914\pi\)
0.749933 0.661513i \(-0.230086\pi\)
\(258\) 0 0
\(259\) 4.00000 + 6.32456i 0.248548 + 0.392989i
\(260\) 0 0
\(261\) 0 0
\(262\) 12.6491 0.781465
\(263\) −29.2843 −1.80575 −0.902873 0.429908i \(-0.858546\pi\)
−0.902873 + 0.429908i \(0.858546\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 10.7967 + 17.0711i 0.661988 + 1.04669i
\(267\) 0 0
\(268\) 0 0
\(269\) −27.9179 −1.70219 −0.851093 0.525014i \(-0.824060\pi\)
−0.851093 + 0.525014i \(0.824060\pi\)
\(270\) 0 0
\(271\) 7.09185i 0.430799i 0.976526 + 0.215400i \(0.0691054\pi\)
−0.976526 + 0.215400i \(0.930895\pi\)
\(272\) 2.23607i 0.135582i
\(273\) 0 0
\(274\) −12.1421 −0.733533
\(275\) 0 0
\(276\) 0 0
\(277\) 21.2132i 1.27458i −0.770625 0.637289i \(-0.780056\pi\)
0.770625 0.637289i \(-0.219944\pi\)
\(278\) 10.7967i 0.647543i
\(279\) 0 0
\(280\) 0 0
\(281\) 28.4853i 1.69929i −0.527356 0.849645i \(-0.676816\pi\)
0.527356 0.849645i \(-0.323184\pi\)
\(282\) 0 0
\(283\) 7.63441 0.453819 0.226909 0.973916i \(-0.427138\pi\)
0.226909 + 0.973916i \(0.427138\pi\)
\(284\) 15.6569i 0.929063i
\(285\) 0 0
\(286\) 1.30986i 0.0774535i
\(287\) −19.1421 + 12.1065i −1.12992 + 0.714627i
\(288\) 0 0
\(289\) 12.0000 0.705882
\(290\) 0 0
\(291\) 0 0
\(292\) −13.9590 −0.816887
\(293\) 25.2982i 1.47794i −0.673740 0.738969i \(-0.735313\pi\)
0.673740 0.738969i \(-0.264687\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.82843i 0.164399i
\(297\) 0 0
\(298\) 0.757359i 0.0438726i
\(299\) 0.926210 0.0535641
\(300\) 0 0
\(301\) −5.07107 8.01806i −0.292291 0.462153i
\(302\) 14.1421 0.813788
\(303\) 0 0
\(304\) 7.63441i 0.437864i
\(305\) 0 0
\(306\) 0 0
\(307\) −3.70484 −0.211446 −0.105723 0.994396i \(-0.533716\pi\)
−0.105723 + 0.994396i \(0.533716\pi\)
\(308\) −2.00000 3.16228i −0.113961 0.180187i
\(309\) 0 0
\(310\) 0 0
\(311\) 5.23943 0.297101 0.148550 0.988905i \(-0.452539\pi\)
0.148550 + 0.988905i \(0.452539\pi\)
\(312\) 0 0
\(313\) −16.5787 −0.937083 −0.468541 0.883442i \(-0.655220\pi\)
−0.468541 + 0.883442i \(0.655220\pi\)
\(314\) −10.7967 −0.609293
\(315\) 0 0
\(316\) 13.0711 0.735305
\(317\) 30.0711 1.68896 0.844480 0.535588i \(-0.179910\pi\)
0.844480 + 0.535588i \(0.179910\pi\)
\(318\) 0 0
\(319\) 1.07107 0.0599683
\(320\) 0 0
\(321\) 0 0
\(322\) 2.23607 1.41421i 0.124611 0.0788110i
\(323\) 17.0711 0.949860
\(324\) 0 0
\(325\) 0 0
\(326\) 7.92893i 0.439143i
\(327\) 0 0
\(328\) 8.56062 0.472681
\(329\) −1.85242 2.92893i −0.102127 0.161477i
\(330\) 0 0
\(331\) 7.92893 0.435814 0.217907 0.975970i \(-0.430077\pi\)
0.217907 + 0.975970i \(0.430077\pi\)
\(332\) 14.3426i 0.787153i
\(333\) 0 0
\(334\) 12.1065i 0.662441i
\(335\) 0 0
\(336\) 0 0
\(337\) 11.9706i 0.652078i 0.945356 + 0.326039i \(0.105714\pi\)
−0.945356 + 0.326039i \(0.894286\pi\)
\(338\) 12.1421 0.660445
\(339\) 0 0
\(340\) 0 0
\(341\) −5.78199 −0.313113
\(342\) 0 0
\(343\) −2.23607 18.3848i −0.120736 0.992685i
\(344\) 3.58579i 0.193333i
\(345\) 0 0
\(346\) 10.2541i 0.551265i
\(347\) 9.07107 0.486960 0.243480 0.969906i \(-0.421711\pi\)
0.243480 + 0.969906i \(0.421711\pi\)
\(348\) 0 0
\(349\) 6.16564i 0.330039i 0.986290 + 0.165020i \(0.0527688\pi\)
−0.986290 + 0.165020i \(0.947231\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.41421i 0.0753778i
\(353\) 18.9737i 1.00987i −0.863158 0.504933i \(-0.831517\pi\)
0.863158 0.504933i \(-0.168483\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.61972 −0.138845
\(357\) 0 0
\(358\) 11.3137i 0.597948i
\(359\) 16.3137i 0.861005i 0.902589 + 0.430502i \(0.141664\pi\)
−0.902589 + 0.430502i \(0.858336\pi\)
\(360\) 0 0
\(361\) −39.2843 −2.06759
\(362\) 2.61972i 0.137689i
\(363\) 0 0
\(364\) −2.07107 + 1.30986i −0.108553 + 0.0686552i
\(365\) 0 0
\(366\) 0 0
\(367\) −24.5967 −1.28394 −0.641970 0.766730i \(-0.721883\pi\)
−0.641970 + 0.766730i \(0.721883\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) 0 0
\(371\) −18.0475 + 11.4142i −0.936977 + 0.592596i
\(372\) 0 0
\(373\) 7.07107i 0.366126i 0.983101 + 0.183063i \(0.0586012\pi\)
−0.983101 + 0.183063i \(0.941399\pi\)
\(374\) −3.16228 −0.163517
\(375\) 0 0
\(376\) 1.30986i 0.0675508i
\(377\) 0.701474i 0.0361277i
\(378\) 0 0
\(379\) −22.0711 −1.13371 −0.566857 0.823816i \(-0.691841\pi\)
−0.566857 + 0.823816i \(0.691841\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0.656854i 0.0336076i
\(383\) 16.0361i 0.819408i −0.912219 0.409704i \(-0.865632\pi\)
0.912219 0.409704i \(-0.134368\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.48528i 0.431889i
\(387\) 0 0
\(388\) 0.542561 0.0275444
\(389\) 22.8284i 1.15745i −0.815524 0.578724i \(-0.803551\pi\)
0.815524 0.578724i \(-0.196449\pi\)
\(390\) 0 0
\(391\) 2.23607i 0.113083i
\(392\) −3.00000 + 6.32456i −0.151523 + 0.319438i
\(393\) 0 0
\(394\) −5.92893 −0.298695
\(395\) 0 0
\(396\) 0 0
\(397\) −29.6114 −1.48616 −0.743078 0.669205i \(-0.766635\pi\)
−0.743078 + 0.669205i \(0.766635\pi\)
\(398\) 21.5934i 1.08238i
\(399\) 0 0
\(400\) 0 0
\(401\) 9.79899i 0.489338i 0.969607 + 0.244669i \(0.0786793\pi\)
−0.969607 + 0.244669i \(0.921321\pi\)
\(402\) 0 0
\(403\) 3.78680i 0.188634i
\(404\) −12.1065 −0.602323
\(405\) 0 0
\(406\) −1.07107 1.69351i −0.0531562 0.0840473i
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) 22.9032i 1.13249i 0.824236 + 0.566246i \(0.191605\pi\)
−0.824236 + 0.566246i \(0.808395\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −10.4130 −0.513014
\(413\) 16.2132 10.2541i 0.797800 0.504573i
\(414\) 0 0
\(415\) 0 0
\(416\) 0.926210 0.0454112
\(417\) 0 0
\(418\) 10.7967 0.528083
\(419\) 28.8441 1.40913 0.704564 0.709640i \(-0.251142\pi\)
0.704564 + 0.709640i \(0.251142\pi\)
\(420\) 0 0
\(421\) 33.3553 1.62564 0.812820 0.582515i \(-0.197931\pi\)
0.812820 + 0.582515i \(0.197931\pi\)
\(422\) −16.2132 −0.789246
\(423\) 0 0
\(424\) 8.07107 0.391966
\(425\) 0 0
\(426\) 0 0
\(427\) −1.30986 2.07107i −0.0633885 0.100226i
\(428\) 2.00000 0.0966736
\(429\) 0 0
\(430\) 0 0
\(431\) 4.79899i 0.231159i −0.993298 0.115580i \(-0.963127\pi\)
0.993298 0.115580i \(-0.0368725\pi\)
\(432\) 0 0
\(433\) −9.71157 −0.466708 −0.233354 0.972392i \(-0.574970\pi\)
−0.233354 + 0.972392i \(0.574970\pi\)
\(434\) 5.78199 + 9.14214i 0.277545 + 0.438837i
\(435\) 0 0
\(436\) 7.07107 0.338643
\(437\) 7.63441i 0.365204i
\(438\) 0 0
\(439\) 4.08849i 0.195133i −0.995229 0.0975664i \(-0.968894\pi\)
0.995229 0.0975664i \(-0.0311058\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.07107i 0.0985106i
\(443\) 23.0711 1.09614 0.548070 0.836433i \(-0.315363\pi\)
0.548070 + 0.836433i \(0.315363\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 13.0328 0.617119
\(447\) 0 0
\(448\) 2.23607 1.41421i 0.105644 0.0668153i
\(449\) 18.3848i 0.867631i −0.901002 0.433816i \(-0.857167\pi\)
0.901002 0.433816i \(-0.142833\pi\)
\(450\) 0 0
\(451\) 12.1065i 0.570075i
\(452\) −1.07107 −0.0503788
\(453\) 0 0
\(454\) 25.1393i 1.17985i
\(455\) 0 0
\(456\) 0 0
\(457\) 17.8284i 0.833979i −0.908911 0.416989i \(-0.863085\pi\)
0.908911 0.416989i \(-0.136915\pi\)
\(458\) 10.7967i 0.504496i
\(459\) 0 0
\(460\) 0 0
\(461\) 38.9394 1.81359 0.906794 0.421575i \(-0.138523\pi\)
0.906794 + 0.421575i \(0.138523\pi\)
\(462\) 0 0
\(463\) 38.2843i 1.77922i 0.456720 + 0.889610i \(0.349024\pi\)
−0.456720 + 0.889610i \(0.650976\pi\)
\(464\) 0.757359i 0.0351595i
\(465\) 0 0
\(466\) −23.0711 −1.06875
\(467\) 29.6114i 1.37025i 0.728424 + 0.685127i \(0.240253\pi\)
−0.728424 + 0.685127i \(0.759747\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −7.25077 −0.333744
\(473\) −5.07107 −0.233168
\(474\) 0 0
\(475\) 0 0
\(476\) 3.16228 + 5.00000i 0.144943 + 0.229175i
\(477\) 0 0
\(478\) 7.17157i 0.328020i
\(479\) 33.6999 1.53979 0.769895 0.638171i \(-0.220309\pi\)
0.769895 + 0.638171i \(0.220309\pi\)
\(480\) 0 0
\(481\) 2.61972i 0.119449i
\(482\) 4.47214i 0.203700i
\(483\) 0 0
\(484\) 9.00000 0.409091
\(485\) 0 0
\(486\) 0 0
\(487\) 29.8995i 1.35488i −0.735580 0.677438i \(-0.763090\pi\)
0.735580 0.677438i \(-0.236910\pi\)
\(488\) 0.926210i 0.0419275i
\(489\) 0 0
\(490\) 0 0
\(491\) 21.5147i 0.970946i −0.874252 0.485473i \(-0.838647\pi\)
0.874252 0.485473i \(-0.161353\pi\)
\(492\) 0 0
\(493\) −1.69351 −0.0762717
\(494\) 7.07107i 0.318142i
\(495\) 0 0
\(496\) 4.08849i 0.183579i
\(497\) −22.1421 35.0098i −0.993211 1.57040i
\(498\) 0 0
\(499\) 22.0711 0.988037 0.494018 0.869451i \(-0.335528\pi\)
0.494018 + 0.869451i \(0.335528\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 18.8148 0.839744
\(503\) 21.8181i 0.972822i −0.873730 0.486411i \(-0.838306\pi\)
0.873730 0.486411i \(-0.161694\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.41421i 0.0628695i
\(507\) 0 0
\(508\) 10.0000i 0.443678i
\(509\) 0.224736 0.00996125 0.00498063 0.999988i \(-0.498415\pi\)
0.00498063 + 0.999988i \(0.498415\pi\)
\(510\) 0 0
\(511\) −31.2132 + 19.7410i −1.38079 + 0.873289i
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 21.2097i 0.935521i
\(515\) 0 0
\(516\) 0 0
\(517\) −1.85242 −0.0814693
\(518\) −4.00000 6.32456i −0.175750 0.277885i
\(519\) 0 0
\(520\) 0 0
\(521\) −17.5049 −0.766903 −0.383452 0.923561i \(-0.625265\pi\)
−0.383452 + 0.923561i \(0.625265\pi\)
\(522\) 0 0
\(523\) 28.4605 1.24449 0.622245 0.782822i \(-0.286221\pi\)
0.622245 + 0.782822i \(0.286221\pi\)
\(524\) −12.6491 −0.552579
\(525\) 0 0
\(526\) 29.2843 1.27685
\(527\) 9.14214 0.398238
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) −10.7967 17.0711i −0.468096 0.740125i
\(533\) 7.92893 0.343440
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 27.9179 1.20363
\(539\) −8.94427 4.24264i −0.385257 0.182743i
\(540\) 0 0
\(541\) 5.07107 0.218022 0.109011 0.994041i \(-0.465232\pi\)
0.109011 + 0.994041i \(0.465232\pi\)
\(542\) 7.09185i 0.304621i
\(543\) 0 0
\(544\) 2.23607i 0.0958706i
\(545\) 0 0
\(546\) 0 0
\(547\) 16.2132i 0.693227i 0.938008 + 0.346613i \(0.112668\pi\)
−0.938008 + 0.346613i \(0.887332\pi\)
\(548\) 12.1421 0.518686
\(549\) 0 0
\(550\) 0 0
\(551\) 5.78199 0.246321
\(552\) 0 0
\(553\) 29.2278 18.4853i 1.24289 0.786074i
\(554\) 21.2132i 0.901263i
\(555\) 0 0
\(556\) 10.7967i 0.457882i
\(557\) −26.1421 −1.10768 −0.553839 0.832624i \(-0.686838\pi\)
−0.553839 + 0.832624i \(0.686838\pi\)
\(558\) 0 0
\(559\) 3.32119i 0.140471i
\(560\) 0 0
\(561\) 0 0
\(562\) 28.4853i 1.20158i
\(563\) 24.3720i 1.02716i 0.858042 + 0.513579i \(0.171681\pi\)
−0.858042 + 0.513579i \(0.828319\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −7.63441 −0.320898
\(567\) 0 0
\(568\) 15.6569i 0.656947i
\(569\) 22.5269i 0.944377i 0.881498 + 0.472189i \(0.156536\pi\)
−0.881498 + 0.472189i \(0.843464\pi\)
\(570\) 0 0
\(571\) −38.2132 −1.59917 −0.799586 0.600551i \(-0.794948\pi\)
−0.799586 + 0.600551i \(0.794948\pi\)
\(572\) 1.30986i 0.0547679i
\(573\) 0 0
\(574\) 19.1421 12.1065i 0.798977 0.505318i
\(575\) 0 0
\(576\) 0 0
\(577\) −23.4458 −0.976062 −0.488031 0.872826i \(-0.662285\pi\)
−0.488031 + 0.872826i \(0.662285\pi\)
\(578\) −12.0000 −0.499134
\(579\) 0 0
\(580\) 0 0
\(581\) 20.2835 + 32.0711i 0.841502 + 1.33053i
\(582\) 0 0
\(583\) 11.4142i 0.472728i
\(584\) 13.9590 0.577626
\(585\) 0 0
\(586\) 25.2982i 1.04506i
\(587\) 15.4277i 0.636771i 0.947961 + 0.318385i \(0.103141\pi\)
−0.947961 + 0.318385i \(0.896859\pi\)
\(588\) 0 0
\(589\) −31.2132 −1.28612
\(590\) 0 0
\(591\) 0 0
\(592\) 2.82843i 0.116248i
\(593\) 46.8916i 1.92561i −0.270202 0.962804i \(-0.587090\pi\)
0.270202 0.962804i \(-0.412910\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.757359i 0.0310226i
\(597\) 0 0
\(598\) −0.926210 −0.0378755
\(599\) 3.68629i 0.150618i 0.997160 + 0.0753089i \(0.0239943\pi\)
−0.997160 + 0.0753089i \(0.976006\pi\)
\(600\) 0 0
\(601\) 12.1065i 0.493836i 0.969036 + 0.246918i \(0.0794179\pi\)
−0.969036 + 0.246918i \(0.920582\pi\)
\(602\) 5.07107 + 8.01806i 0.206681 + 0.326792i
\(603\) 0 0
\(604\) −14.1421 −0.575435
\(605\) 0 0
\(606\) 0 0
\(607\) 43.9541 1.78404 0.892020 0.451996i \(-0.149288\pi\)
0.892020 + 0.451996i \(0.149288\pi\)
\(608\) 7.63441i 0.309616i
\(609\) 0 0
\(610\) 0 0
\(611\) 1.21320i 0.0490810i
\(612\) 0 0
\(613\) 15.8579i 0.640493i 0.947334 + 0.320247i \(0.103766\pi\)
−0.947334 + 0.320247i \(0.896234\pi\)
\(614\) 3.70484 0.149515
\(615\) 0 0
\(616\) 2.00000 + 3.16228i 0.0805823 + 0.127412i
\(617\) 32.0000 1.28827 0.644136 0.764911i \(-0.277217\pi\)
0.644136 + 0.764911i \(0.277217\pi\)
\(618\) 0 0
\(619\) 18.2064i 0.731776i 0.930659 + 0.365888i \(0.119235\pi\)
−0.930659 + 0.365888i \(0.880765\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −5.23943 −0.210082
\(623\) −5.85786 + 3.70484i −0.234690 + 0.148431i
\(624\) 0 0
\(625\) 0 0
\(626\) 16.5787 0.662618
\(627\) 0 0
\(628\) 10.7967 0.430835
\(629\) −6.32456 −0.252177
\(630\) 0 0
\(631\) 10.2843 0.409410 0.204705 0.978824i \(-0.434376\pi\)
0.204705 + 0.978824i \(0.434376\pi\)
\(632\) −13.0711 −0.519939
\(633\) 0 0
\(634\) −30.0711 −1.19427
\(635\) 0 0
\(636\) 0 0
\(637\) −2.77863 + 5.85786i −0.110093 + 0.232097i
\(638\) −1.07107 −0.0424040
\(639\) 0 0
\(640\) 0 0
\(641\) 28.4853i 1.12510i 0.826763 + 0.562550i \(0.190180\pi\)
−0.826763 + 0.562550i \(0.809820\pi\)
\(642\) 0 0
\(643\) 29.9951 1.18289 0.591446 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(644\) −2.23607 + 1.41421i −0.0881134 + 0.0557278i
\(645\) 0 0
\(646\) −17.0711 −0.671652
\(647\) 37.0869i 1.45804i 0.684493 + 0.729019i \(0.260023\pi\)
−0.684493 + 0.729019i \(0.739977\pi\)
\(648\) 0 0
\(649\) 10.2541i 0.402510i
\(650\) 0 0
\(651\) 0 0
\(652\) 7.92893i 0.310521i
\(653\) 8.14214 0.318626 0.159313 0.987228i \(-0.449072\pi\)
0.159313 + 0.987228i \(0.449072\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −8.56062 −0.334236
\(657\) 0 0
\(658\) 1.85242 + 2.92893i 0.0722148 + 0.114182i
\(659\) 48.3848i 1.88480i 0.334483 + 0.942402i \(0.391438\pi\)
−0.334483 + 0.942402i \(0.608562\pi\)
\(660\) 0 0
\(661\) 18.2064i 0.708146i 0.935218 + 0.354073i \(0.115204\pi\)
−0.935218 + 0.354073i \(0.884796\pi\)
\(662\) −7.92893 −0.308167
\(663\) 0 0
\(664\) 14.3426i 0.556602i
\(665\) 0 0
\(666\) 0 0
\(667\) 0.757359i 0.0293251i
\(668\) 12.1065i 0.468416i
\(669\) 0 0
\(670\) 0 0
\(671\) −1.30986 −0.0505665
\(672\) 0 0
\(673\) 34.7990i 1.34140i 0.741728 + 0.670701i \(0.234007\pi\)
−0.741728 + 0.670701i \(0.765993\pi\)
\(674\) 11.9706i 0.461089i
\(675\) 0 0
\(676\) −12.1421 −0.467005
\(677\) 22.9032i 0.880243i −0.897938 0.440122i \(-0.854935\pi\)
0.897938 0.440122i \(-0.145065\pi\)
\(678\) 0 0
\(679\) 1.21320 0.767297i 0.0465585 0.0294462i
\(680\) 0 0
\(681\) 0 0
\(682\) 5.78199 0.221404
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 2.23607 + 18.3848i 0.0853735 + 0.701934i
\(687\) 0 0
\(688\) 3.58579i 0.136707i
\(689\) 7.47550 0.284794
\(690\) 0 0
\(691\) 23.9884i 0.912560i 0.889836 + 0.456280i \(0.150819\pi\)
−0.889836 + 0.456280i \(0.849181\pi\)
\(692\) 10.2541i 0.389804i
\(693\) 0 0
\(694\) −9.07107 −0.344333
\(695\) 0 0
\(696\) 0 0
\(697\) 19.1421i 0.725060i
\(698\) 6.16564i 0.233373i
\(699\) 0 0
\(700\) 0 0
\(701\) 41.8701i 1.58141i −0.612197 0.790705i \(-0.709714\pi\)
0.612197 0.790705i \(-0.290286\pi\)
\(702\) 0 0
\(703\) 21.5934 0.814410
\(704\) 1.41421i 0.0533002i
\(705\) 0 0
\(706\) 18.9737i 0.714083i
\(707\) −27.0711 + 17.1212i −1.01811 + 0.643911i
\(708\) 0 0
\(709\) −24.1421 −0.906677 −0.453338 0.891338i \(-0.649767\pi\)
−0.453338 + 0.891338i \(0.649767\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.61972 0.0981780
\(713\) 4.08849i 0.153115i
\(714\) 0 0
\(715\) 0 0
\(716\) 11.3137i 0.422813i
\(717\) 0 0
\(718\) 16.3137i 0.608822i
\(719\) 23.9884 0.894615 0.447307 0.894380i \(-0.352383\pi\)
0.447307 + 0.894380i \(0.352383\pi\)
\(720\) 0 0
\(721\) −23.2843 + 14.7263i −0.867152 + 0.548435i
\(722\) 39.2843 1.46201
\(723\) 0 0
\(724\) 2.61972i 0.0973610i
\(725\) 0 0
\(726\) 0 0
\(727\) −6.70820 −0.248794 −0.124397 0.992233i \(-0.539700\pi\)
−0.124397 + 0.992233i \(0.539700\pi\)
\(728\) 2.07107 1.30986i 0.0767589 0.0485466i
\(729\) 0 0
\(730\) 0 0
\(731\) 8.01806 0.296559
\(732\) 0 0
\(733\) 39.6408 1.46417 0.732084 0.681214i \(-0.238548\pi\)
0.732084 + 0.681214i \(0.238548\pi\)
\(734\) 24.5967 0.907883
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 0 0
\(739\) 26.3553 0.969497 0.484748 0.874654i \(-0.338911\pi\)
0.484748 + 0.874654i \(0.338911\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 18.0475 11.4142i 0.662543 0.419029i
\(743\) 35.1421 1.28924 0.644620 0.764503i \(-0.277016\pi\)
0.644620 + 0.764503i \(0.277016\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 7.07107i 0.258890i
\(747\) 0 0
\(748\) 3.16228 0.115624
\(749\) 4.47214 2.82843i 0.163408 0.103348i
\(750\) 0 0
\(751\) 31.2132 1.13899 0.569493 0.821996i \(-0.307140\pi\)
0.569493 + 0.821996i \(0.307140\pi\)
\(752\) 1.30986i 0.0477656i
\(753\) 0 0
\(754\) 0.701474i 0.0255462i
\(755\) 0 0
\(756\) 0 0
\(757\) 44.0416i 1.60072i 0.599520 + 0.800360i \(0.295358\pi\)
−0.599520 + 0.800360i \(0.704642\pi\)
\(758\) 22.0711 0.801657
\(759\) 0 0
\(760\) 0 0
\(761\) 31.3050 1.13480 0.567402 0.823441i \(-0.307949\pi\)
0.567402 + 0.823441i \(0.307949\pi\)
\(762\) 0 0
\(763\) 15.8114 10.0000i 0.572411 0.362024i
\(764\) 0.656854i 0.0237642i
\(765\) 0 0
\(766\) 16.0361i 0.579409i
\(767\) −6.71573 −0.242491
\(768\) 0 0
\(769\) 15.0441i 0.542504i 0.962508 + 0.271252i \(0.0874376\pi\)
−0.962508 + 0.271252i \(0.912562\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.48528i 0.305392i
\(773\) 17.3460i 0.623892i −0.950100 0.311946i \(-0.899019\pi\)
0.950100 0.311946i \(-0.100981\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −0.542561 −0.0194768
\(777\) 0 0
\(778\) 22.8284i 0.818439i
\(779\) 65.3553i 2.34160i
\(780\) 0 0
\(781\) −22.1421 −0.792308
\(782\) 2.23607i 0.0799616i
\(783\) 0 0
\(784\) 3.00000 6.32456i 0.107143 0.225877i
\(785\) 0 0
\(786\) 0 0
\(787\) 33.9247 1.20928 0.604642 0.796497i \(-0.293316\pi\)
0.604642 + 0.796497i \(0.293316\pi\)
\(788\) 5.92893 0.211209
\(789\) 0 0
\(790\) 0 0
\(791\) −2.39498 + 1.51472i −0.0851557 + 0.0538572i
\(792\) 0 0
\(793\) 0.857864i 0.0304637i
\(794\) 29.6114 1.05087
\(795\) 0 0
\(796\) 21.5934i 0.765357i
\(797\) 11.0214i 0.390399i −0.980764 0.195199i \(-0.937465\pi\)
0.980764 0.195199i \(-0.0625354\pi\)
\(798\) 0 0
\(799\) 2.92893 0.103618
\(800\) 0 0
\(801\) 0 0
\(802\) 9.79899i 0.346014i
\(803\)